Research ArticleAccelerated Degradation Process Analysis Based on theNonlinear Wiener Process with Covariates and Random Effects

Li Sun1 Xiaohui Gu1 and Pu Song2

1Nanjing University of Science and Technology Nanjing Jiangsu 210094 China2Science and Technology on Combustion and Explosion Laboratory Xirsquoan Shanxi 710065 China

Correspondence should be addressed to Xiaohui Gu gxiaohuinjusteducn

Received 13 September 2016 Accepted 27 November 2016

Academic Editor Eusebio Valero

Copyright copy 2016 Li Sun et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

It is assumed that the drift parameter is dependent on the acceleration variables and the diffusion coefficient remains the sameacross the whole accelerated degradation test (ADT) in most of the literature based on Wiener process However the diffusioncoefficient variation would also become obvious in some applications with the stress increasing Aiming at the phenomenon thepaper concludes that both the drift parameter and the diffusion parameter depend on stress variables based on the invarianceprinciple of failure mechanism and Nelson assumption Accordingly constant stress accelerated degradation process (CSADP) andstep stress accelerated degradation process (SSADP) with random effects are modeledThe unknown parameters in the establishedmodel are estimated based on the property of degradation and degradation increment separately for CASDT and SSADT by themaximum likelihood estimation approach with measurement error In addition the simulation steps of accelerated degradationdata are provided and simulated step stress accelerated degradation data is designed to validate the proposed model compared toother models Finally a case study of CSADT is conducted to demonstrate the benefits of our model in the practical engineering

1 Introduction

For many highly reliable products it is not an easy taskto obtain their life information by using traditional life testbecause failures are not likely to occur in a certain periodof time even by censoring life test and accelerated life testIn such a case degradation data which is related to life isused due to the following reasons ease of obtaining lowcost short test period and informative data And it has beenwidely used in classification [1] residual life estimation [2]reliability assessment [3] and so on To model the degra-dation data two classes of models have been well exploitedgeneral path model and stochastic process model [4 5] Thegeneral path model is first introduced by Lu and Meekerin 1993 [6] whose failure time is determined with knownrandom parameters But it may not be good at describing theinherent randomness of each product and the unexplainedrandomness and dynamics due to unobserved environ-mental factors There are various stochastic process mod-els including Wiener process [7] Gamma process [8] Ge-ometric Brownian Motion Process [9] and Inverse Gaussian

Process [10 11] Beyond all of the stochastic process modelsWiener process has been used intensively for its flexible andmeaningful characteristic In addition Wiener process hasmore advantages than other stochastic process models fornonmonotonic degradation data

Most of the degradation data mentioned above is degra-dation data under normal stress or field degradation dataHowever the life information should be obtained in a shorterperiod of time for some products especially for newly devel-oped products and highly reliable components Instead it isa lengthy and drawn-out process to collect field degradationdata Under the circumstances ADT is a suitable choice togather the life information quickly and efficiently

In general with more accelerated degradation data andhigher measuring precision we can achieve higher accuracyfor forecasting parameters but the experiment cost wouldincrease correspondingly So we can deal with the optimalaccelerated degradation plan (including the optimal settingsfor the sample size accelerated stresses measurement fre-quency and termination time) for a Wiener degradationprocess by minimizing the approximate variance of the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 5246108 13 pageshttpdxdoiorg10115520165246108

2 Mathematical Problems in Engineering

estimated mean time to failure under the constraint thatthe total experimental cost does not exceed a prespecifiedbudget or minimizing the testing cost under the conditionof a maximum acceptable approximate standard error Somewell-known references on the optimization of CSADT basedon Wiener process are Lim and Yum [5] and Tsai et al [12]The optimization of SSADT based on Wiener process can bereferred to in the research of Liao and Tseng [13] Tang et al[14] and Hu et al [15] In addition accelerated degradationmodel is another hot area which has attractedmuch attentionof the researchers Liao and Tseng [13]modeled the step stressaccelerated degradation data of LED lamps Consideringunit-to-unit variability a step stress accelerated degradationmodel based on the basic Wiener process was proposed byTang [16]However it is often found that the degradation pathis not always linear So an accelerated degradation processmodeling method with random effects for the nonlinearWiener process was established by Tang et al [17] later Wanget al [18] proposed a Bayesian evaluationmethod to integratethe ADT data from laboratory with the failure data fromfield

The above literature all assumed that the drift parameteris dependent on the acceleration variables and the diffusioncoefficient remains the same across the whole ADT Butwhen the degradation rate increases the degradation vari-ation would also become larger in some applications [19]WHITMORE [20] fitted the degradation data of each productseparately with a time scale transformedWiener process andthen the parameter transformation was tentatively identifiedbased on the plots against the reciprocal of the absolutetemperature The plots revealed that both the drift parameterand the diffusion parameter of self-regulating heating cableare increasing with the increment of temperature Doksumand Hoyland [21] introduced the conception of the multi-plicative factor and it was assumed that the drift parameterand diffusion parameter are multiples of the multiplicativefactor whose expression with accelerated stress level providesa good model fit of one of the four empirical models Liaoand Elsayed [22] extended an accelerated degradation modelto predict field reliability by considering the stress variationswhere the drift parameter and the diffusion parameter areexpressed by the different function of the constant stressvector Ye [19] proposed a new random effectsWiener processmodel such that the drift parameter is a particular multipleof the diffusion parameter and the unknown parameterswere calculated by EM algorithm The relationship betweenthe drift parameter and diffusion parameter was either anassumption or just the fitting according to specific test data

The paper is motivated by the latest paper of Wang et al[23] in which they deduced that the ratio of drift parametersunder two different stresses is equal to the accelerationfactor as well as the ratio of diffusion parameters Basedon this conclusion we model the constant stress accelerateddegradation process (CSADP) and the step stress accelerateddegradation process (SSADP) in consideration of randomeffects Moreover the unknown parameters in the modelincluding measurement error are obtained by using themaximum likelihood estimation (MLE) method Besides anumerical example and a case study are presented to verify

the superiority of themodel proposed in this paper comparedwith other two models

The remainder of this paper is organized as followsSection 2 develops the nonlinearWiener process and deducesthe relationships of parameters in ADT and the probabilitydensity function (PDF) and cumulative distribution function(CDF) under a certain stress with random effects Section 3models the degradation process in CSADT and SSADT Sec-tion 4 describes the procedure for parameter estimation fortwo cases Two numerical examples and a practical exampleare presented to verify the proposed model in Sections 5 and6 separately Section 7 concludes the paper with a discussion

2 Nonlinear Wiener Process with Covariatesand Random Effects

21 The Wiener Process with Time Scale TransformationThe time-transformed Wiener process is commonly used tomodel the nonlinear accelerated degradation data [17] Let119883(119905) denote the degradation value at time 119905 then the Wienerdegradation process with time scale transformation can berepresented as follows [20]119883(119905) = 120582Λ (119905) + 120590119861119861 (Λ (119905)) (1)

where 120582 is the drift parameter 120590119861 is the diffusion parameter119905 denotes the clock or calendar time and Λ(119905) is the trans-formed time whose selection can be referred to in Section 6of literature [24] 119861(Λ(119905)) is the standard Brownian motionwhich represents the stochastic dynamics of the degradationprocess at transformed time scale If Λ(119905) = 119905 the nonlinearWiener process becomes the traditional Wiener process [17]Generally if 119883(119905) reaches a specific value 119908 which is relatedto the failure mechanism in most cases for the first time theproduct is announced to be failed and the time is thus calledthe first hitting time (FHT) Given 119908 120582 and 120590119861 it is knownthat the transformed FHT in such a case follows an inverseGaussian distribution [11] with corresponding PDF and CDFas 119891 (119905)

= 119908radic21205871205902119861 (Λ (119905))3 exp(minus(119908 minus 120582Λ (119905))221205902119861Λ (119905) ) 119889Λ (119905)119889119905 (2)

119865 (119905)= Φ(120582Λ (119905) minus 119908radic1205902119861Λ (119905) )

+ exp(21205821199081205902119861 )Φ(minus120582Λ (119905) + 119908radic1205902119861Λ (119905) ) (3)

where Φ(sdot) denotes a standard normal distribution function

22 Deducing the Relationship of Parameters in ADT Based onNonlinearWiener Process ADT is a method to accelerate the

Mathematical Problems in Engineering 3

Table 1 Three accelerated models and their acceleration factor

Accelerated models Drift parameter 120582 Diffusion parameter 120590119861 Acceleration factor

Arrhenius model 120582119896 = 120578 exp(minus 120573119878119896 ) (1205902119861)119896 = 120581 exp(minus 120573119878119896 ) 1205721198960 = exp(minus120573( 1119878119896 minus 11198780 ))Inverse power model 120582119896 = 120578 exp (minus120573 ln(119878119896)) (1205902119861)119896 = 120581 exp (minus120573 ln(119878119896)) 1205721198960 = exp (minus120573(ln(119878119896) minus ln(1198780)))Eyring model 120582119896 = 120578119878119896 exp (minus120573119878119896) (1205902119861)119896 = 120581119878119896 exp (minus120573119878119896) 1205721198960 = 1198780119878119896 exp (minus120573 (1119878119896 minus 11198780))degradation of products by elevating stress and the obtaineddegradation data are then used to extrapolate the informationthrough accelerating model to obtain the estimates of life orperformance of products at normal use condition To ensurethe accuracy of the extrapolation the failure mechanismunder the accelerated stress and the normal stress mustkeep the same which is also the premise of the ADT Oneof the most common methods for consistency inspectionof the failure mechanism is based on statistical method[25] The principle of this method is that the accelerationfactor is a constant and independent of testing time if thefailure mechanism remains unchanged The definition of theacceleration factor is given below according to the Nelsonassumption [26]

Specify 120596 = Λ(119905) and suppose 119865lowast represents thepredetermined cumulative failure probability 1205960 is definedas the testing time when the accumulated failure probabilitycomes to 119865lowast under normal stress 1198780 as well as 120596119896 underaccelerated stress 119878119896

119865lowast = 119865119896 (120596119896) = 1198650 (1205960) (4)

Then the acceleration factor 1205721198960 of stress 119878119896 relative tostress 1198780 can be defined as

1205721198960 = 1205960120596119896 (5)

The expression 1205960 = 1205721198960120596119896 can be obtained from (5)and plug it into (4) Then take the first-order derivative withrespect to 120596119896 and we have the following equation for any120596119896 gt 0

119891119896 (120596119896) = 12057211989601198910 (1205721198960120596119896) (6)

The expression of 119891119896(119905119896) and 1198910(1199050) can be deductedaccording to (2) then

1205721198960 = 119891119896 (120596119896)1198910 (1205721198960120596119896) = radic (1205721198960)3 (1205902119861)0(1205902119861)119896sdot exp[ 12120596119896 ( 1199082(1205902119861)0 1205721198960 minus 1199082(1205902119861)119896)+ 1205961198962 (120582201205721198960(1205902119861)0 minus 1205822119896(1205902119861)119896) + 119908( 120582119896(1205902119861)119896 minus 1205820(1205902119861)0)]

(7)

The acceleration factor 1205721198960 is a constant that does notchangewith 119905119896 if and only if the relationship of the parametersis satisfied with

1205721198960 = 1205821198961205820 = (1205902119861)119896(1205902119861)0 (8)

Instead of the hypothesis that the diffusion parameter is aconstant and does not change with the stress the conclusionthat both the drift parameter 120582 and the diffusion parameter120590119861 depend on stress variables could be drawn based on theprevious derivation At the same time it was testified thata unit with high realization of the drift parameter wouldpossess a high degradation rate and a high variation in thedegradation path in theory which is in conformity with theviewpoint of [19] The relationship between parameters andaccelerated stress variables can be set up by accelerationmodels based on engineering background The frequentlyused acceleration model includes the Arrhenius model theinverse power model and the Eyring model whose expres-sions and acceleration factors are listed in Table 1 Specify120589(119878119896 | 120573) = exp(minus120573119878119896) 120589(119878119896 | 120573) = exp(minus120573 ln(119878119896)) and120589(119878119896 | 120573) = exp(minus120573119878119896)119878119896 to the three models separatelyThe accelerated model of drift parameter 120582 the diffusionparameter120590119861 and the accelerated factor1205721198960 can be uniformlywritten as

120582119896 = 120578120589119896 (9)

(1205902119861)119896 = 120581120589119896 (10)

1205721198960 = 1205891198961205890 (11)

where 120589119896 is the abbreviated form of 120589(119878119896 | 120573) for simplificationof the expressions

23 ADT with Random Effects The observed degradation forproducts from the same population may be very differentowing to unobservable factors [4 27] And there are threesources of variability contributing to the nondeterminism ofdegradation (1) temporal variability (2) unit-to-unite vari-ability and (3) measurement variability [28] The temporalvariability is referred to as the inherent stochastic character-istics of the standard Brownian motion The measurementvariability is usually consideredwhile the degradation param-eters were estimated The unit-to-unite variability is usuallymodeled as random effects of the degradation parameterIt is very difficult to model the accelerated processes if the

4 Mathematical Problems in Engineering

drift parameter 120582 and the diffusion parameter 120590119861 are bothconsidered as the randomparameters So like Peng and Tseng[29] Si et al [30 31] and Tsai et al [32] it is also assumedthat different units have different drift parameters while alldiffusion parameters have the same value under a certain

stress Then we have 120578 sim 119873(120583120578 1205902120578) where 120583120578 and 1205902120578 are themean and variance of the parameter 120578 separately

Considering the random effects and the effects of accel-erated stresses on the drift parameter 120582 and the diffusionparameter 120590119861 we have the PDF and CDF of FHT under stress119878119896 as

119891119878119896 (119905) = 119908radic21205871205963 (120581120589119896 + 12059021205781205892119896120596) exp(minus (119908 minus 120583120578120589119896120596)22120596 (120581120589119896 + 12059021205781205892119896120596)) 119889120596119889119905 (12)

119865119878119896 (119905) =Φ( 120583120578120589119896120596 minus 119908radic120596(120581120589119896 + 12059021205781205892119896120596)) + exp(2120583120578119908120581 + 2120590212057811990821205812 ) sdot Φ(minus21205902120578120589119896119908120596 + 120581 (120583120578120589119896120596 + 119908)120581radic120581120589119896120596 + 120590212057812058921198961205962 ) if 119905 lt infinΦ(120583120578120590120578) + exp(2120583120578119908120581 + 2120590212057811990821205812 )Φ(minus21205902120578119908 + 120581120583120578120581120590120578 ) if 119905 997888rarr infin

(13)

whereΦ(sdot) is the distribution function of the standard normaldistribution

When 120583120578120590120578 approach infinity the transformed FHTunder stress 119878119896 is subject to inverse Gaussian (IG) distribu-tion

3 Model the CSADP and SSADP withRandom Effects

As themost-usedADTCSADTand SSADThave beenwidelyresearched But the models are quite different while the driftparameter 120582 and the diffusion parameter 120590119861 all depend onstress variables especially for SSADT

31 Modeling the CSADP with Random Effects Let 1198781 119878119896 119878119870 denote119870 stress level higher than normal stresslevel 1198780 such that 1198781 lt sdot sdot sdot lt 119878119870 and the subscript 119896 iscorresponding to the 119896th stress level 119896 = 1 2 119870 Supposethat there are 119873119896 units of samples tested under a constantaccelerated stress 119878119896 and each sample is measured 119872119896119894 timesat the 119896th stress level 119894 = 1 2 119873119896 The degradationat transformed time 120596119896119894119895 where 120596119896119894119895 = Λ(119905119896119894119895) is 119909119896119894119895 119895 =1 2 119872119896119894 The degradation process can be formulated as

119909119896119894119895 (120596119896119894119895 | 119878119896) = 120578119894120589119896120596119896119894119895 + radic120581120589119896119861 (120596119896119894119895) (14)

32Modeling the SSADPwith RandomEffects Similarly withCSADT it was assumed that there are 119870 accelerated stresslevels 1198781 lt sdot sdot sdot 119878119896 lt sdot sdot sdot lt 119878119870 in the whole test But thenumber of the samples is only 119873 which is different fromsum119870119896=1119873119896 inCSADT that verifies the characteristic that SSADTneeds fewer samples compared with CSADT [33] Supposethat each sample is also measured 119872119896119894 times at the 119896thstress level 119894 = 1 2 119873 119896 = 1 2 119870 And 120591119896119894 is thetransformed time scale where 120591119896119894 = Λ(119905119896119894119872119896119894) at which thestress changes from the 119896th stress level to the (119896 + 1)th stresslevel of the 119894th sample 119896 = 1 2 119870 minus 1 Besides 1205911198940 equals

0 and 120591119894119870 is the transformed end time of the 119870-step stressaccelerated test of the 119894th sample

The degradation process for CSADT is the same asSSADT under accelerated stress 1198781

1199091119894119895 = 12057811989412058911205961119894119895 + radic1205811205891119861 (1205961119894119895) 0 le 1205961119894119895 le 1205911119894 (15)

Raise the accelerated stress up to 1198782 at transformed timepoint 1205911119894 for the 119894th product Then the degradation is drivenby the accelerated stress 1198782 and the corresponding parameterscan be obtained by (9) are (10) where the drift parameter andthe diffusion parameter are equal to 1205781198941205892 and 1205811205892 separatelyThus the degradation consisted of the degradation driven by1198781 and the degradation driven by 1198782

1199092119894119895 = 12057811989412058911205911198941 + radic1205811205891119861 (1205911198941) + 1205781198941205892 (1205961198942119895 minus 1205911198941)+ radic1205811205892119861 (1205961198942119895 minus 1205911198941) (16)

Similarly the accelerated stress is turning up to 1198783 attransformed timepoint 1205912119894 and the degradation can bewrittenas

1199093119894119895 = 2sum119899=1

[120578119894120589119899 (120591119894119899 minus 120591119894(119899minus1)) + radic120581120589119899119861 (120591119894119899 minus 120591119894(119899minus1))]+ 1205781198941205893 (1205961198943119895 minus 1205911198942) + radic1205811205893119861 (1205961198943119895 minus 1205911198942)

(17)

Mathematical Problems in Engineering 5

According to the analysis the degradation process ofSSADT can be formulated as

119909119896119894119895 =

1205781198941205891120596119894119896119895 + radic1205811205891119861 (120596119894119896119895) 1205911198940 le 120596119894119896119895 le 120591119894112057811989412058911205911198941 + radic1205811205891119861 (1205911198941) + 1205781198941205892 (120596119894119896119895 minus 1205911198941) + radic1205811205892119861 (120596119894119896119895 minus 1205911198941) 1205911198941 le 120596119894119896119895 le 1205911198942sdot sdot sdot119896minus1sum119899=1

[120578119894120589119899 (120591119894119899 minus 120591119894(119899minus1)) + radic120581120589119899119861 (120591119894119899 minus 120591119894(119899minus1))] + 120578119894120589119896 (120596119894119896119895 minus 1205911198942) + radic120581120589119896119861 (120596119894119896119895 minus 1205911198942) 120591119894119896minus1 le 120596119894119896119895 le 120591119894119896sdot sdot sdot119870minus1sum119899=1

[120578119894120589119899 (120591119894119899 minus 120591119894(119899minus1)) + 120581120589119899119861 (120591119894119899 minus 120591119894(119899minus1))] + 120578119894120589119870 (120596119894119896119895 minus 1205911198942) + 120581120589119870119861 (120596119894119896119895 minus 1205911198942) 120591119894119870minus1 le 120596119894119896119895 le 120591119894119870

(18)

4 Parameter Estimation

In real applications it is inevitable that some measurementerrors may be introduced during the observation process[34] When a measurement is taken the observed degrada-tion is shown as follows for both CSADT and SSADT

119910119896119894119895 (120596119896119894119895 | 119878119896) = 119909119896119894119895 (120596119896119894119895 | 119878119896) + 120576119896119894119895119896 = 1 119870 119894 = 1 119873119896 119895 = 1 119872119896119894 (19)

where the measurement errors 120576119896119894119895 are assumed to be iidrealizations of 120576119896119894119895 sim 119873(0 1205902120576 ) and mutually independent of119909119896119894119895 [7 34] and 119873119896 is equal to 119873 for all of the acceleratedstresses of SSADT

The unknown parameters in the models are Θ =120583120578 1205902120578 120581 120573 1205902120576 There are two methods to deal with theunknown parameters with each considering one of thetwo main properties of Brownian motion which could becharacterized as follows [35]

(P1) The increment Δ119909(120596) = 119909(120596 + Δ120596) minus 119909(120596) isindependent of the degradation 119909(120596) which meansthat if 0 le 1205961 le 1205962 le 1205963 le 1205964 then 119909(1205962) minus119909(1205961) and 119909(1205964) minus 119909(1205963) are independent randomvariables Meanwhile the increment Δ119909(120596) follows anormal distribution where the mean equals 120582Δ120596 andthe variance is 1205902119861Δ120596

(P2) According to (P1) the degradation119909(120596) follows a nor-mal distribution with the mean 120582120596 and the variance1205902119861120596

Even though our models concern the unit-to-unitevariability the essence of the Wiener process remains thesame Owing to space constraints this paper deals with theunknown parameter based on property (P2) in the case ofCSADT and property (P1) in the case of SSADT

41 Parameter Estimation of CSADT Specify 119879119896119894 = (1198791198961198941 119879119896119894119872119896119894)1015840 119879119896119894119895 = 120596119896119894119895 119884119896119894 = (1198841198961198941 119884119896119894119872119896119894)1015840 119884119896 =(1198841198961 119884119896119873119896) and 119884 = (1198841 119884119870) for 119896 = 1 119870 119894 =

1 119873119896 and 119895 = 1 119872119896119894 According to property (P1) andindependent assumption of Brownian motion 119884119896119894 follows amultivariate normal distribution with mean and variance

119896119894 = 120583120578120589119896119879119896119894Σ119896119894 = 1205902120578Σ119896119894 (20)

where

Σ119896119894 = Ω119896119894 + 1205892119896119879119896119894119879119879119896119894119876119896119894 =

[[[[[[[[

1205961198961198941 1205961198961198941 sdot sdot sdot 12059611989611989411205961198961198941 1205961198961198942 sdot sdot sdot 1205961198961198942 d1205961198961198941 1205961198961198942 sdot sdot sdot 120596119896119894119872119896119894

]]]]]]]]

Ω119896119894 = 120589119896119876119896119894 + 2120576119868119896119894

(21)

and 119868119896119894 is an identified matrix of order119872119896119894The log-likelihood function of unknown parameters Θ =120583120578 1205902120578 120573 2120576 is

ln 119871 (Θ | 119884)= minus12 ln (2120587) 119870sum

119896=1

119873119896sum119894=1

119872119894119896 minus 12 ln (1205902120578) 119870sum119896=1

119873119896sum119894=1

119872119896119894minus 12 119870sum119896=1

119873119896sum119894=1

10038161003816100381610038161003816Σ11989611989410038161003816100381610038161003816minus 121205902120578

119870sum119896=1

119873119896sum119894=1

(119910119896119894 minus 120583120578120589119896119879119896119894)1015840 Σminus1119896119894 (119910119896119894 minus 120583120578120589119896119879119896119894)

(22)

6 Mathematical Problems in Engineering

Taking the first partial derivatives of the log-likelihoodfunction with respect to 120583120578 1205902120578 yields

120597 ln 119871 (Θ | 119884)120597120583120578= 11205902120578 (

119870sum119896=1

119873119896sum119894=1

1205891198961198791015840119896119894Σminus1119896119894 119910119896119894 minus 120583120578 119870sum119896=1

119873119896sum119894=1

12058921198961198791015840119896119894Σminus1119896119894 119879119896119894) 120597 ln 119871 (Θ | 119884)1205971205902120578 = minus 121205902120578

119870sum119896=1

119873119896sum119894=1

119872119896119894 + 12 (1205902120578)2sdot 119870sum119896=1

119873119896sum119894=1

(119910119896119894 minus 120583120578120589119896119879119896119894)1015840 Σminus1119896119894 (119910119896119894 minus 120583120578120589119896119879119896119894)

(23)

For the special value of ( 120573 2120576) setting the derivation of120597 ln 119871(Θ | 119884)120597120583120578 120597 ln 119871(Θ | 119884)1205971205902120578 to zero the MLE for120583120578 1205902120578 can be expressed as

120578 = sum119870119896=1sum119873119896119894=1 1205891198961198791015840119896119894Σminus1119896119894 119910119896119894sum119870119896=1sum119873119896119894=1 12058921198961198791015840119896119894Σminus1119896119894 119879119896119894 2120578 = 1sum119870119896=1sum119873119896119894=1119872119896119894

sdot 119870sum119896=1

119873119896sum119894=1

(119910119896119894 minus 120578120589119896119879119896119894)1015840 Σminus1119896119894 (119910119896119894 minus 120578120589119896119879119896119894) (24)

Substituting (24) into (22) and simplifying the profilelog-likelihood function can be written as

ln 119871 (Θ | 119884) = minus12 (ln (2120587) + 1) 119870sum119896=1

119873119896sum119894=1

119872119894119896minus 12 ln (2120578) 119870sum

119896=1

119873119896sum119894=1

119872119896119894minus 12 119870sum119896=1

119873119896sum119894=1

10038161003816100381610038161003816Σ11989611989410038161003816100381610038161003816 (25)

The MLE of 120573 and 2120576 can be obtained by maximizingthe profile log-likelihood function in (25) through a three-dimensional search Then substitute them into (24) we canobtain MLE for 120583120578 and 1205902120578

The value of and 2120576 can be obtained by the followingequations

= 120581 sdot 21205782120576 = 1205902120576 sdot 2120578 (26)

42 Parameter Estimation of SSADT The degradation pro-cess of SSADT shown as (18) is relatively complicated Thuswe introduce the accelerated factor first for the sake ofsimplicity The variable 120577119896 is represented by the product of 1205771and the accelerated factor 1205721198961

120589119896 = 12058911205721198961 (27)

The SSADP can be rewritten as

119909119896119894119895

=

12057811989412058911205961119894119895 + radic1205811205891119861 (12059612119895) 1205911198940 le 120596119894119896119895 le 12059111989411205781198941205891 (1205911119894 + 12057221 (1205962119894119895 minus 1205911119894)) + radic1205811205891119861 (1205911119894 + 12057221 (1205962119894119895 minus 1205911119894)) 1205911119894 le 1205962119894119895 le 1205912119894sdot sdot sdot1205781198941205891(119896minus1sum

119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119896119894119895 minus 120591(119896minus1)119894)) + radic1205811205891119861(119896minus1sum119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119896119894119895 minus 120591(119896minus1)119894)) 120591(119896minus1)119894 le 120596119896119894119895 le 120591119896119894sdot sdot sdot1205781198941205891(119870minus1sum

119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119870119894119895 minus 120591(119870minus1)119894)) + radic1205811205891119861(119870minus1sum119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119870119894119895 minus 120591(119870minus1)119894)) 120591(119870minus1)119894 le 120596119870119894119895 le 120591119870119894

(28)

Specify 120594119894119896119895 = sum119896minus1119899=1 1205721198991(120591119894119899 minus 120591119894(119899minus1)) + 12057221(120596119894119896119895 minus 120591119894(119896minus1))then the SSADP can be expressed a general formula asfollows

119909119896119894119895 = 1205781198941205891120594119896119894119895 + radic1205811205891119861 (120594119896119894119895) 120591(119896minus1)119894 le 120596119896119894119895 ge 120596119896119894119896 = 1 119870 119894 = 1 119873 119895 = 1 119872119896119894(29)

The foregoing transformation is equivalent to convertingthe degradation driven by stress 119878119896 to the degradation understress 1198781 in physics and the converted equivalent time is 120594119896119894119895At the moment the matrix of degradation and equivalenttransformed time are still bidimensional Because of thecontinuity of the degradation process the matrix can bewritten as a column vector with119872119894 element where119872119894 equalsthe sum of119872119894119896 119896 = 1 119870 and the subscript can be written

Mathematical Problems in Engineering 7

as 119898 for the 119894th product 119898 = 1 119872119894 Then the observeddegradation can be expressed as

119910119894119898 = 119909119894119898 + 120576119894119898119909119894119898 = 1205781198941205891120594119894119898 + 1205811205891119861 (120594119894119898) 1 le 119898 le 119870sum

119896=1

119872119894119896(30)

Define Δ1205941198941 = 1205941198941 Δ120594119894119898 = 120594119894119898 minus 120594119894(119898minus1) Δ120594119894 =(Δ1205941198941 Δ120594119894119872119894)1015840 119877119894 = Δ120594119894 and Δ119910119894119898 = 119910119894119898 minus 119910119894(119898minus1) for119894 = 1 119873 119898 = 1 119872119894 Then Δ119910119894 = (Δ1199101198941 Δ119910119894119872119894)1015840follows a multivariate normal distribution 119873(1205831205781205891119877119894 Π119894)where

Π119894 = 119882119894 + 1205902120578120589211198771198941198771015840119894 (119882119901119902)119894 = cov (Δ119910119894119901 Δ119910119894119902)

=

1205811205891119877119894119901 + 1205902120576 119901 = 119902 = 11205811205891119877119894119901 + 21205902120576 119901 = 119902 gt 1minus1205902120576 119901 = 119902 + 1 or 119901 = 119902 minus 10 otherwise

(31)

So the log-likelihood function can be expressed as

ln 119871 (Θ | 119884)= minus12 ln (2120587) 119873sum

119894=1

119872119894 minus 12 119873sum119894=1

ln (1003816100381610038161003816Π1198941003816100381610038161003816)minus 12 119873sum119894=1

(Δ119910119894 minus 1205831205781205891119877119894)1015840Πminus1119894 (Δ119910119894 minus 1205831205781205891119877119894) (32)

where1003816100381610038161003816Π1198941003816100381610038161003816 = 10038161003816100381610038161198821198941003816100381610038161003816 (1 + 1205902120578120589211198771015840119894119882minus1119894 119877119894)Πminus1119894 = 119882minus1119894 minus 1205902120578120589211 + 1205902120578120589211198771015840119894119882minus1119894 119877119894119882minus1119894 1198771198941198771015840119894119882minus1119894 (33)

Taking the first partial derivatives of the log-likelihoodfunction with respect to 120583120578 1205902120578 yields

120597 ln 119871 (Θ | 119884)120597120583120578 = 1205891 119873sum119894=1

1198771015840119894Πminus1119894 Δ119910119894 minus 12058312057812058921 119873sum119894=1

1198771015840119894Πminus1119894 119877119894 (34)

120597 ln 119871 (Θ | 119884)1205971205902120578 = minus 119873sum119894=1

120589211198771015840119894119882minus1119894 1198771 + 1205902120578120589211198771015840119894119882minus1119894 119877119894+ 119873sum119894=1

12058921 (Δ119910119894 minus 1205831205781205891119877119894)1015840119882minus1119894 1198771198941198771015840119894119882minus1119894 (Δ119910119894 minus 1205831205781205891119877119894)(1 + 1205902120578120589211198771015840119894119882minus1119894 119877)2(35)

For the special value of (120581 120573 1205902120576 ) setting the derivation of120597 ln 119871(Θ | 119884)120597120583120578 to zero the restricted MLE for 120583120578 can beexpressed as

120578 = sum119873119894=1 1198771015840119894Πminus1119894 Δ1199101198941205891sum119873119894=1 1198771015840119894Πminus1119894 119877119894 (36)

Similarly set the derivation of 120597 ln 119871(Θ | 119884)120597120583120578 to zeroand the results of the MLE for 1205902120578 will be discussed on twocases

Case 1 It was assumed that the number of the measurementsand the measurement points of each sample are the same forall of the samples under all of the accelerated stressThat is tosay the subscript of 119877119894 Π119894 and119882119894 can be removed

Thus the restricted MLE for 120583120578 can be expressed as

120578 = sum119873119894=1 1198771015840Πminus1119894 Δ11991011989412058911198731198771015840Πminus1119894 119877 (37)

The first partial derivatives of the log-likelihood functionto 1205902120578 can be rewritten as

120597 ln 119871 (Θ | 119884)1205971205902120578 = minus119873 120589211198771015840119882minus11198771 + 1205902120578120589211198771015840119882minus1119877+ 12058921(1 + 1205902120578120589211198771015840119882minus1119877)2sdot 119873sum119894=1

(Δ119910119894 minus 1205831205781205891119877)119882minus11198771198771015840119882minus1 (Δ119910119894 minus 1205831205781205891119877)1015840 (38)

For the special value of (120581 120573 1205902120576 ) setting the derivation of120597 ln 119871(Θ | 119884)1205971205902120578 to zero the result of the MLE for 1205902120578 can beexpressed as2120578

= sum119873119894=1 (Δ119910119894 minus 1205831205781205891119877)119882minus11198771198771015840119882minus1 (Δ119910119894 minus 1205831205781205891119877)101584011987312058921 (1198771015840119882minus1119877)2minus 1120589211198771015840119882minus1119877

(39)

Substituting (37) (39) into (32) and simplifying the profilelog-likelihood function can be written as

ln 119871 (120581 120573 1205902120576 | 120578 2120578 119884) = minus1198721198732 ln (2120587) minus 1198732 minus 1198732sdot ln (|119882|)minus 12

119873sum119894=1

Δ1199101015840119894119882minus1Δ119910119894 minus sum119873119894=1 (1198771015840119882minus1Δ119910119894)21198771015840119882minus1119877 minus 12sdot ln

sum119873119894=1 (1198771015840119882minus1Δ119910119894)21198731198771015840119882minus1119877 minus sum119873119894=1 (1198771015840119882minus1Δ119910119894)211987321198771015840119882minus1119877

(40)

8 Mathematical Problems in Engineering

where119872 is themeasurement time point of each samplewhichis the same for all of the samples based on the assumption ofCase 1

The MLE of 120581 120573 and 1205902120576 can be obtained by maximizingthe profile log-likelihood function in (40) through a three-dimensional searchThen substitute them into (37) and (39)we can obtain the MLE for 120583120578 and 1205902120578 Case 2 The number of the measurements and the measure-ment points of each sample are different for all of the samplesunder all of the accelerated stress In this case the first partialderivatives of the log-likelihood function to 1205902120578 are shown asin (35)There may be no analytical form by setting the partialderivatives to zero The evaluation of 120583120578 can be expressed as(36) Then the profile log-likelihood function of 1205902120578 120581 120573 1205902120576can be written as

ln 119871 (120581 120573 120590120576 1205902120578 | 120578 119884)= minus12 ln (2120587) 119873sum

119894=1

119872119894 minus 12 119873sum119894=1

ln (1003816100381610038161003816Π1198941003816100381610038161003816)minus 119873sum119894=1

(Δ119910119894 minus 1205781205891119877119894)Πminus1119894 (Δ119910119894 minus 1205781205891119877119894)1015840 (41)

The MLE of 1205902120578 120581 120573 and 1205902120576 can be obtained by maxi-mizing the profile log-likelihood function in (41) through afour-dimensional search Then substitute them into (36) wecan obtain the MLE for 120583120578

It is not to say that we can only use degradation forCSADT and increment for SSADT but just make an intro-duction to both of the two methods in the limited space Inaddition we could verify the results by comparing the esti-mation calculated by the two methods to avoid computationerrors

5 Simulation Data Analysis

In order to validate the model described before and theparameter estimation methods simulation test was con-ducted

51 The Simulation Method of CSADT Data The parametersΘ = 120583120578 1205902120578 120581 120573 1205902120576 should be given before the simulationand simulation process is shown as in the following steps

Step 1 Set 119896 = 1Step 2 Generate 119873119894 random numbers with subscripts 119894 119894 =1 2 119873119896 subject to normal distribution with mean 120583120578 andvariance 1205902120578 Step 3 Let 119897 = 0Step 4 Calculate119883119896119894(119897+1) using the Euler approximation [36]

119883119896119894(119897+1) = 119883119896119894119897 + 120578119894120589119896120596 (Δ119905) + 120581120589119896radic120596 (Δ119905)Ψ (42)

where Ψ is the random number according to the standardnormal distribution and Δ119905 is the step size in simulation and1198831198961198940 = 0 As such set 119897 = 119897 + 1 and return to Step 4 to get thedegradation path until 119897 = 119871119896119894+1 where 119871119896119894 is the simulationsteps for 119894th under stress 119878119896Step 5 Set 119896 = 119896 + 1 and return to Step 2 until 119896 = 119870 + 1Step 6 Extract the degradation based on the predefinedmeasurement time point 119905119896119894119895 where 119896 = 1 2 119870 119894 =1 2 119873119896 and 119895 = 1 2 119872119896119894 from the data set 119883119896119894119897 andget the data set of degradation119883119896119894119895Step 7 Simulate the measured degradation 119884119896119894119895 = 119883119896119894119895 + 120576119896119894119895where 120576119896119894119895 sim 119873(0 1205902120576 )52 The Simulation Method of SSADT Data There are somedifferences in the simulation process for SSADT comparedwith CSADT The simulation process is shown as follows

Step 1 Generate 119873 random numbers with subscripts 119894 119894 =1 2 119873 subject to normal distribution with mean 120583120578 andvariance 1205902120578 Step 2 Let 119896 = 1Step 3 Set 119897 = 1Step 4 Calculate119883119896119894(119897+1)

119883119896119894(119897+1) = 119883119896119894119897 + 120578119894120589119896120596 (Δ119905) + 120581120589119896radic120596 (Δ119905)Ψ (43)

where Ψ is the random number according to the standardnormal distribution and Δ119905 is the step size in simulation and1198831198961198940 = 01198831198961198940 = 119883(119896minus1)119894119871(119896minus1) (119896 gt 1) As such set 119897 = 119897+1 andreturn to Step 4 to get the degradation path until 119897 = 119871119896119894 + 1Step 5 Set 119896 = 119896 + 1 and return to Step 2 until 119896 = 119870 + 1Step 6 Extract the degradation based on the predefinedmeasurement time point 119905119896119894119895 where 119896 = 1 2 119870 119894 =1 2 119873 and 119895 = 1 2 119872119894119896 from the data set 119883119896119894119897 andget the data set of degradation119883119896119894119895Step 7 Simulate the measured degradation 119884119896119894119895 = 119883119896119894119895 + 120576119896119894119895where 120576119896119894119895 sim 119873(0 1205902120576 )53 The Analysis of the Simulated SSADT Data We justgive the analysis of simulated SSADT data here becausewe would give a case study of the CSADT later It wasassumed that the accelerated stress is temperature and thesimulation test contains 4 stresseswhich are 50∘C 60∘C 70∘Cand 80∘C For simplicity the transformed time function isset as Λ(119905) = 119905 and 20 degradation paths are generatedThen the parameters for degradation process are assumedas Θ = 5 4 025 minus3000 001 Moreover the time intervalmeasurement is 25 h for each sample of the whole simulationtest with 40 measurement time points The degradation dataare depicted as in Figure 1

Mathematical Problems in Engineering 9

Table 2 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 log-LF AIC RETruth value 5 1 025 mdash minus3000 001 mdash mdash mdash1198720 501 139 028 mdash minus297561 00078 142228 minus283456 084021198721 512 175 mdash 12907 lowast 10minus4 minus298081 00129 112289 minus223577 562711198722 620 mdash mdash 15213 lowast 10minus4 minus305175 13789 lowast 10minus9 109361 minus217721 78049

0 200 400 600 800 1000

0

05

1

15

2

25

th

y

Figure 1 The simulation degradation paths of SSADT

For simplicity the degradation model for SSADT pro-posed in this paper is referred to as1198720 the model presentedby Tang et al [17] as 1198721 and the stochastic SSADT modeldescribed in reference [13] as 1198722 The estimation results ofthe unknown parameters the log-likelihood function valueand the Akaike information criterion (AIC) [37] which isevaluated by (44) are summarized in Table 2 As we cansee from Table 2 our model clearly outperforms model interms of the log-LF and AIC The estimates of 120583120578 1205902120578 and1205902120576 are more close to the truth values compared with 1198721and 1198722 Meanwhile the diffusion-related variables includeparameters 120581 and 120573 and the diffusion parameter under thefour stresses can be obtained by (10) which is (28377 lowast 10minus548534 lowast 10minus5 78244 lowast 10minus5 12001 lowast 10minus4) in 1198721 and(22334lowast10minus5 39912lowast10minus5 64597lowast10minus5 99423lowast10minus5) forthe true model Moreover taking 1198781 as a reference a criterionformulizing the relative error (RE) is shown as (45) It is clearthat1198720 has the smallest RE from Table 2

AIC (119873) = minus2 ln 119871 (Θ) + 2119873 (44)

RE = 1003816100381610038161003816100381610038161003816100381610038161003816120583120578 minus 120578120578

1003816100381610038161003816100381610038161003816100381610038161003816 +100381610038161003816100381610038161003816100381610038161003816100381610038161205902120578 minus 21205782120578

10038161003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161003816(1205902119861)1 minus (2119861)1(2119861)1

10038161003816100381610038161003816100381610038161003816100381610038161003816+ 1003816100381610038161003816100381610038161003816100381610038161003816120573 minus

1003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161205902120576 minus 21205762120576

1003816100381610038161003816100381610038161003816100381610038161003816 (45)

It is assumed that the failure threshold119908 is equal to 5 andthe normal stress is 25∘C The PDF and CDF under normalstress are shown as in Figure 2 The phenomenon can beobserved where the PDF and CDF of our model most nearlyapproached the real model By contrasting 1198721 with 1198722 itcan be concluded that the neglect of random effects wouldresult in sharper PDF which is consistent with the result inreference [17] And it is noteworthy that the 05 quantiles ofthe failure life are almost equal from the CDF of the threemodels But there is a greater difference between the realmodel and 1198721 while the indicator is taken as mean time tofailure as well as the difference between the real model and1198722 So the correlation between the stress and the diffusionparameter should not be neglected Next we are going toprove the superiority of our model in practical application

6 Case Application

The CSADT model with covariates and random effects isverified by the accelerated degradation data of carbon-filmresistors whose raw data set is explicitly given in Table C3of Meeker and Escobar [38]There were 29 samples and threeaccelerated temperature stresses in the whole test where ninesamples were observed at 83∘C the remaining two stresses133∘C and 173∘C had 10 samples for each stress All of thesamples were observed at the same time points with 1199050 = 01199051 = 452 1199052 = 1030 1199053 = 4341 and 1199054 = 8084 (inhours) Similar to [39] it was assumed that the standardoperating temperature was 50∘C and the threshold value forpercent increase in resistance was taken to be 119862 = 12 Itis reasonable and computationally easier to use the ratios inresistance to the initial value for each rather than the percentincrease [9] For ease of calculation the logarithm of the ratiowas taken as the degradation here That is the value 028of the percent increase was changed to 2796119864 minus 3 and soon and the transformed threshold value 119908 was 01133 Thedegradation paths of all the sample were as shown in Figure 3It can be seen that the degradation of the samples uniformlyshowed a nonlinear characteristic especially at the beginningof the ADT Thus the degradation is modeled with the timescale transformedWiener process as (1) with 120596 = Λ(119905) = 119905119887

Similarly the proposed degradation model for SSADT inthis paper is referred to as1198720 themodel presented by Tang etal [17] as1198721 and themodel described in reference [9] as1198723The estimation results of the unknown parameters the log-likelihood function value and the AIC are shown in Table 3The estimated parameters 119887 are approximately equal for thethree models Take 119887 = 050 as an example to show the time

10 Mathematical Problems in Engineering

Table 3 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 b Log-LF AIC MTTF1198720 823 207 108119864 minus 02 mdash minus420282 182119864 minus 20 050 51848 minus102497 380119864 + 071198721 1471 433 mdash 413119864 minus 07 minus458656 506119864 minus 20 053 48637 minus96073 515119864 + 071198722 1111 mdash mdash 676119864 minus 07 minus447982 411119864 minus 18 053 46837 minus92674 406119864 + 07

0 1 2 3 4 50

05

1

15

2

25

th

The P

DF

Real model

times104

times10minus4

M1

M0

M2

0 1 2 3 4 50

02

04

06

08

1

th

The C

DF

times104

Real modelM1

M0

M2

Figure 2 The comparison of the PDF and CDF of the three models for simulated SSADT data

0 2000 4000 6000 80000

0002

0004

0006

0008

001

th

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 2000 4000 6000 80000

001

002

003

th

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 2000 4000 6000 8000th

0

002

004

006

008

01

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 3 The degradation paths of carbon-film resistors

scale transformed degradation paths as in Figure 4 which areapproximately straight lines compared with Figure 3

Our model has the largest log-LF and smallest AICcompared with1198721 and1198722 The result proves that our modelis more suitable for the degradation data Besides 1198721 hasthe larger log-LF and smaller AIC compared with1198722 So therandom effects could not be neglected It can be observed thatthe variance of the drift parameter is slightly largerwhichmaybe owing to the fixed value of diffusion parameter in1198721

The PDF and CDF under the standard operating temper-ature are as shown in Figure 5The time corresponding to thepeak values of PDF were as follows 2676 lowast 107 1049 lowast 107

and 5028lowast106 for1198720 to1198722 But theMTTF of1198720 isminimalfrom Table 3 And the uncertainty in the estimated PDFs ofthe lifetimes under1198720 is smallest compared to1198721 and1198722 asseen in Figure 5When time 119905 approaches positive infinity theCDF of ourmodel which is calculated by (13) tends illimitablyto 1 and the CDF of 1198721 is roughly equal to 09997 It followsthat our model could do well in distinguishing the differencefrom random effects and the effect of covariates Anotherthing which is worth noting is that the measurement error isquite tiny in the case but it is just an individual phenomenonThemeasurement error should also be considered in practicalapplication

Mathematical Problems in Engineering 11

0 20 40 60 800

0002

0004

0006

0008

001

Λ(t)

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 20 40 60 800

001

002

003

Λ(t)

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 20 40 60 800

002

004

006

008

01

Λ(t)

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 4 The degradation paths of carbon-film resistors under transformed time scale

0 2 4 6 8 100

05

1

15

2

25

3

35

th

times10minus8

M0

M1

M2

The P

DF

times107

0 2 4 6 8 100

02

04

06

08

1

th

M0

M1

M2

The C

DF

times107

Figure 5 The comparison of the PDF and CDF of the three models for real CSADT data

7 Conclusion

In this paper the degradation models based on nonlinearWiener process are established for both constant stress accel-erated degradation data and step stress accelerated degrada-tion data Before the establishment the relationship betweenthe drift parameter and stress variables is derived based onthe invariance principle of failure mechanism and Nelsonassumption so is the relationship between the diffusionparameter and stress variables It is concluded that the ratioof drift parameters under two stresses is a constant which isirrelevant to the testing time and depends only on the twostresses as long as the ratio of diffusion parameters is equalto the ratio of drift parameters And the ratio is defined asaccelerated factor Besides the random effects are also takeninto consideration where the drift parameter is assumed to be

normally distributed and the diffusion parameter is same forall of the samples under a certain stress Then the PDF andCDF of the FHT are deduced considering random effects

Because of the dependency between the diffusion param-eter and stress variables the degradation process is quitedifferent either for CSADT or for SSADT The CSADP andSSADP with random effects are modeled Moreover theunknown parameters are solved by MLE based on the twoproperties of Wiener process At the end of the paper thesimulated data of SSADT and the CSADTdata of carbon-filmresistors are both analyzed to verify the proposed model Itis concluded that the model has the biggest log-LF and thesmallest AIC compared with the two other models

The innovation of this paper lies in the following Firstthe random effects are considered under the new relationshipbetween the diffusion parameter and accelerated stresses

12 Mathematical Problems in Engineering

Second the degradation process was modeled for bothCSADT and SSADT Thirdly the unknown parameters wereestimated based on the two properties of Wiener process andthe result of theMLE for 1205902120578 is discussed on two cases Fourththe measurement error of the degradation data is also valued

However we have only considered the random effects ofthe drift parameter in this paper due to the complexity of thecomputation A further research may consider the randomeffects of the diffusion parameter into the model At the sametime the study of the paper may provide new ideas for therelativity analysis between the parameters of other stochasticprocess and stress variables

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by Jiangsu Province GraduateStudent Scientific Research Innovation Project of China(Project KYLX15 0330)The help is gratefully acknowledged

References

[1] H-F Yu ldquoOptimal classification of highly-reliable productswhose degradation paths satisfyWiener processesrdquo EngineeringOptimization vol 35 no 3 pp 313ndash324 2003

[2] XWang N Balakrishnan and B Guo ldquoResidual life estimationbased on a generalized Wiener degradation processrdquo ReliabilityEngineering and System Safety vol 124 pp 13ndash23 2014

[3] Y Ren Q Feng T Ye and B Sun ldquoA novel model of reliabilityassessment for circular electrical connectorsrdquo IEEE Transac-tions on Components Packaging andManufacturing Technologyvol 5 no 6 pp 755ndash761 2015

[4] Z-S Ye and M Xie ldquoStochastic modelling and analysis ofdegradation for highly reliable productsrdquo Applied StochasticModels in Business and Industry vol 31 no 1 pp 16ndash32 2015

[5] H Lim and B-J Yum ldquoOptimal design of accelerated degrada-tion tests based on Wiener process modelsrdquo Journal of AppliedStatistics vol 38 no 2 pp 309ndash325 2011

[6] C J Lu and W Q Meeker ldquoUsing degradation measures toestimate a time-to-failure distributionrdquo Technometrics vol 35no 2 pp 161ndash174 1993

[7] G AWhitmore ldquoEstimating degradation by a wiener diffusionprocess subject to measurement errorrdquo Lifetime Data Analysisvol 1 no 3 pp 307ndash319 1995

[8] Z Pan and N Balakrishnan ldquoReliability modeling of degra-dation of products with multiple performance characteristicsbased on gamma processesrdquo Reliability Engineering amp SystemSafety vol 96 no 8 pp 949ndash957 2011

[9] C Park and W J Padgett ldquoAccelerated degradation modelsfor failure based on geometric Brownian motion and gammaprocessesrdquo Lifetime Data Analysis vol 11 no 4 pp 511ndash5272005

[10] Z-S Ye and N Chen ldquoThe inverse Gaussian process as adegradation modelrdquo Technometrics vol 56 no 3 pp 302ndash3112014

[11] A Onar and W J Padgett ldquoAccelerated test models with theinverse Gaussian distributionrdquo Journal of Statistical Planningand Inference vol 89 no 1-2 pp 119ndash133 2000

[12] T-R Tsai Y L Lio and N Jiang ldquoOptimal decisions on theaccelerated degradation test plan under the Wiener processrdquoQuality Technology and Quantitative Management vol 11 no4 pp 461ndash470 2014

[13] C-M Liao and S-T Tseng ldquoOptimal design for step-stressaccelerated degradation testsrdquo IEEE Transactions on Reliabilityvol 55 no 1 pp 59ndash66 2006

[14] L C Tang G Yang and M Xie ldquoPlanning of step-stress accel-erated degradation testrdquo in Proceedings of the Annual Reliabil-ity and Maintainability SymposiummdashProceedings InternationalSymposium on Product Quality and Integrity pp 287ndash292January 2004

[15] C-H Hu M-Y Lee and J Tang ldquoOptimum step-stress accel-erated degradation test for Wiener degradation process underconstraintsrdquo European Journal of Operational Research vol 241no 2 pp 412ndash421 2015

[16] S Tang ldquoStep stress accelerated degradation process modelingand remaining useful life estimationrdquo Journal of MechanicalEngineering vol 50 no 16 p 33 2014

[17] S Tang X Guo C Yu H Xue and Z Zhou ldquoAccelerated degra-dation tests modeling based on the nonlinear wiener processwith random effectsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 560726 11 pages 2014

[18] L Wang R Pan X Li and T Jiang ldquoA Bayesian reliability eval-uation method with integrated accelerated degradation testingand field informationrdquo Reliability Engineering amp System Safetyvol 112 pp 38ndash47 2013

[19] Z-S Ye N Chen and Y Shen ldquoA new class of Wiener processmodels for degradation analysisrdquo Reliability Engineering ampSystem Safety vol 139 pp 58ndash67 2015

[20] G A Whitmore and F Schenkelberg ldquoModelling accelerateddegradation data using wiener diffusion with a time scaletransformationrdquo Lifetime Data Analysis vol 3 no 1 pp 27ndash451997

[21] K A Doksum and A Hoyland ldquoModels for variable-stressaccelerated life testing experiments based on Wiener processesand the inverse Gaussian distributionrdquo Technometrics vol 34no 1 pp 74ndash82 1992

[22] H Liao and E A Elsayed ldquoReliability inference for field con-ditions from accelerated degradation testingrdquo Naval ResearchLogistics vol 53 no 6 pp 576ndash587 2006

[23] H-WWang T-X Xu andW-YWang ldquoRemaining life predic-tion based on wiener processes with ADT prior informationrdquoQuality and Reliability Engineering International vol 32 no 3pp 753ndash765 2015

[24] C-Y Peng ldquoInverse Gaussian processes with random effectsand explanatory variables for degradation datardquo Technometricsvol 57 no 1 pp 100ndash111 2015

[25] X Pan XHuang Y Chen YWang and R Kang ldquoConnotationof failure mechanism consistency and identificationmethod foraccelerated testingrdquo in Proceedings of the Prognostics and SystemHealth Management Conference (PHM-Shenzhen rsquo11) May 2011

[26] W B Nelson Accelerated Testing Statistical Models Test Plansand Data Analysis John Wiley amp Sons 2009

[27] Z-S Ye Y Hong and Y Xie ldquoHow do heterogeneities in oper-ating environments affect field failure predictions and testplanningrdquo The Annals of Applied Statistics vol 7 no 4 pp2249ndash2271 2013

Mathematical Problems in Engineering 13

[28] Z-X Zhang X-S Si C-H Hu Q Zhang T Li and C XuldquoPlanning repeated degradation testing for products with three-source variabilityrdquo IEEE Transactions on Reliability vol 65 no2 pp 640ndash647 2016

[29] C-Y Peng and S-T Tseng ldquoMis-specification analysis of lineardegradation modelsrdquo IEEE Transactions on Reliability vol 58no 3 pp 444ndash455 2009

[30] X-S Si W Wang C-H Hu D-H Zhou and M G PechtldquoRemaining useful life estimation based on a nonlinear diffu-sion degradation processrdquo IEEE Transactions on Reliability vol61 no 1 pp 50ndash67 2012

[31] X-S Si W Wang C-H Hu M-Y Chen and D-H Zhou ldquoAWiener-process-based degradationmodel with a recursive filteralgorithm for remaining useful life estimationrdquo MechanicalSystems and Signal Processing vol 35 no 1-2 pp 219ndash237 2013

[32] C-C Tsai S-T Tseng and N Balakrishnan ldquoMis-specificationanalyses of gamma andWiener degradation processesrdquo Journalof Statistical Planning and Inference vol 141 no 12 pp 3725ndash3735 2011

[33] F Haghighi and S J Bae ldquoReliability estimation from lineardegradation and failure time data with competing risks under astep-stress accelerated degradation testrdquo IEEE Transactions onReliability vol 64 no 3 pp 960ndash971 2015

[34] Z-S Ye Y Wang K-L Tsui and M Pecht ldquoDegradation dataanalysis usingwiener processeswithmeasurement errorsrdquo IEEETransactions on Reliability vol 62 no 4 pp 772ndash780 2013

[35] J Huang D S Golubovic S Koh et al ldquoDegradation modelingof mid-power white-light LEDs by using Wiener processrdquoOptics Express vol 23 no 15 pp A966ndashA978 2015

[36] A Beskos O Papaspiliopoulos G O Roberts and P Fearn-head ldquoExact and computationally efficient likelihood-basedestimation for discretely observed diffusion processes (withdiscussion)rdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 68 no 3 pp 333ndash382 2006

[37] J Shang and J E Cavanaugh ldquoAn assumption for the develop-ment of bootstrap variants of the Akaike information criterionin mixed modelsrdquo Statistics and Probability Letters vol 78 no12 pp 1422ndash1429 2008

[38] W QMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[39] W J Padgett andM A Tomlinson ldquoInference from accelerateddegradation and failure data based on Gaussian process mod-elsrdquo Lifetime Data Analysis vol 10 no 2 pp 191ndash206 2004

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Mathematical Problems in Engineering

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2 Mathematical Problems in Engineering

estimated mean time to failure under the constraint thatthe total experimental cost does not exceed a prespecifiedbudget or minimizing the testing cost under the conditionof a maximum acceptable approximate standard error Somewell-known references on the optimization of CSADT basedon Wiener process are Lim and Yum [5] and Tsai et al [12]The optimization of SSADT based on Wiener process can bereferred to in the research of Liao and Tseng [13] Tang et al[14] and Hu et al [15] In addition accelerated degradationmodel is another hot area which has attractedmuch attentionof the researchers Liao and Tseng [13]modeled the step stressaccelerated degradation data of LED lamps Consideringunit-to-unit variability a step stress accelerated degradationmodel based on the basic Wiener process was proposed byTang [16]However it is often found that the degradation pathis not always linear So an accelerated degradation processmodeling method with random effects for the nonlinearWiener process was established by Tang et al [17] later Wanget al [18] proposed a Bayesian evaluationmethod to integratethe ADT data from laboratory with the failure data fromfield

The above literature all assumed that the drift parameteris dependent on the acceleration variables and the diffusioncoefficient remains the same across the whole ADT Butwhen the degradation rate increases the degradation vari-ation would also become larger in some applications [19]WHITMORE [20] fitted the degradation data of each productseparately with a time scale transformedWiener process andthen the parameter transformation was tentatively identifiedbased on the plots against the reciprocal of the absolutetemperature The plots revealed that both the drift parameterand the diffusion parameter of self-regulating heating cableare increasing with the increment of temperature Doksumand Hoyland [21] introduced the conception of the multi-plicative factor and it was assumed that the drift parameterand diffusion parameter are multiples of the multiplicativefactor whose expression with accelerated stress level providesa good model fit of one of the four empirical models Liaoand Elsayed [22] extended an accelerated degradation modelto predict field reliability by considering the stress variationswhere the drift parameter and the diffusion parameter areexpressed by the different function of the constant stressvector Ye [19] proposed a new random effectsWiener processmodel such that the drift parameter is a particular multipleof the diffusion parameter and the unknown parameterswere calculated by EM algorithm The relationship betweenthe drift parameter and diffusion parameter was either anassumption or just the fitting according to specific test data

The paper is motivated by the latest paper of Wang et al[23] in which they deduced that the ratio of drift parametersunder two different stresses is equal to the accelerationfactor as well as the ratio of diffusion parameters Basedon this conclusion we model the constant stress accelerateddegradation process (CSADP) and the step stress accelerateddegradation process (SSADP) in consideration of randomeffects Moreover the unknown parameters in the modelincluding measurement error are obtained by using themaximum likelihood estimation (MLE) method Besides anumerical example and a case study are presented to verify

the superiority of themodel proposed in this paper comparedwith other two models

The remainder of this paper is organized as followsSection 2 develops the nonlinearWiener process and deducesthe relationships of parameters in ADT and the probabilitydensity function (PDF) and cumulative distribution function(CDF) under a certain stress with random effects Section 3models the degradation process in CSADT and SSADT Sec-tion 4 describes the procedure for parameter estimation fortwo cases Two numerical examples and a practical exampleare presented to verify the proposed model in Sections 5 and6 separately Section 7 concludes the paper with a discussion

2 Nonlinear Wiener Process with Covariatesand Random Effects

21 The Wiener Process with Time Scale TransformationThe time-transformed Wiener process is commonly used tomodel the nonlinear accelerated degradation data [17] Let119883(119905) denote the degradation value at time 119905 then the Wienerdegradation process with time scale transformation can berepresented as follows [20]119883(119905) = 120582Λ (119905) + 120590119861119861 (Λ (119905)) (1)

where 120582 is the drift parameter 120590119861 is the diffusion parameter119905 denotes the clock or calendar time and Λ(119905) is the trans-formed time whose selection can be referred to in Section 6of literature [24] 119861(Λ(119905)) is the standard Brownian motionwhich represents the stochastic dynamics of the degradationprocess at transformed time scale If Λ(119905) = 119905 the nonlinearWiener process becomes the traditional Wiener process [17]Generally if 119883(119905) reaches a specific value 119908 which is relatedto the failure mechanism in most cases for the first time theproduct is announced to be failed and the time is thus calledthe first hitting time (FHT) Given 119908 120582 and 120590119861 it is knownthat the transformed FHT in such a case follows an inverseGaussian distribution [11] with corresponding PDF and CDFas 119891 (119905)

= 119908radic21205871205902119861 (Λ (119905))3 exp(minus(119908 minus 120582Λ (119905))221205902119861Λ (119905) ) 119889Λ (119905)119889119905 (2)

119865 (119905)= Φ(120582Λ (119905) minus 119908radic1205902119861Λ (119905) )

+ exp(21205821199081205902119861 )Φ(minus120582Λ (119905) + 119908radic1205902119861Λ (119905) ) (3)

where Φ(sdot) denotes a standard normal distribution function

22 Deducing the Relationship of Parameters in ADT Based onNonlinearWiener Process ADT is a method to accelerate the

Mathematical Problems in Engineering 3

Table 1 Three accelerated models and their acceleration factor

Accelerated models Drift parameter 120582 Diffusion parameter 120590119861 Acceleration factor

Arrhenius model 120582119896 = 120578 exp(minus 120573119878119896 ) (1205902119861)119896 = 120581 exp(minus 120573119878119896 ) 1205721198960 = exp(minus120573( 1119878119896 minus 11198780 ))Inverse power model 120582119896 = 120578 exp (minus120573 ln(119878119896)) (1205902119861)119896 = 120581 exp (minus120573 ln(119878119896)) 1205721198960 = exp (minus120573(ln(119878119896) minus ln(1198780)))Eyring model 120582119896 = 120578119878119896 exp (minus120573119878119896) (1205902119861)119896 = 120581119878119896 exp (minus120573119878119896) 1205721198960 = 1198780119878119896 exp (minus120573 (1119878119896 minus 11198780))degradation of products by elevating stress and the obtaineddegradation data are then used to extrapolate the informationthrough accelerating model to obtain the estimates of life orperformance of products at normal use condition To ensurethe accuracy of the extrapolation the failure mechanismunder the accelerated stress and the normal stress mustkeep the same which is also the premise of the ADT Oneof the most common methods for consistency inspectionof the failure mechanism is based on statistical method[25] The principle of this method is that the accelerationfactor is a constant and independent of testing time if thefailure mechanism remains unchanged The definition of theacceleration factor is given below according to the Nelsonassumption [26]

Specify 120596 = Λ(119905) and suppose 119865lowast represents thepredetermined cumulative failure probability 1205960 is definedas the testing time when the accumulated failure probabilitycomes to 119865lowast under normal stress 1198780 as well as 120596119896 underaccelerated stress 119878119896

119865lowast = 119865119896 (120596119896) = 1198650 (1205960) (4)

Then the acceleration factor 1205721198960 of stress 119878119896 relative tostress 1198780 can be defined as

1205721198960 = 1205960120596119896 (5)

The expression 1205960 = 1205721198960120596119896 can be obtained from (5)and plug it into (4) Then take the first-order derivative withrespect to 120596119896 and we have the following equation for any120596119896 gt 0

119891119896 (120596119896) = 12057211989601198910 (1205721198960120596119896) (6)

The expression of 119891119896(119905119896) and 1198910(1199050) can be deductedaccording to (2) then

1205721198960 = 119891119896 (120596119896)1198910 (1205721198960120596119896) = radic (1205721198960)3 (1205902119861)0(1205902119861)119896sdot exp[ 12120596119896 ( 1199082(1205902119861)0 1205721198960 minus 1199082(1205902119861)119896)+ 1205961198962 (120582201205721198960(1205902119861)0 minus 1205822119896(1205902119861)119896) + 119908( 120582119896(1205902119861)119896 minus 1205820(1205902119861)0)]

(7)

The acceleration factor 1205721198960 is a constant that does notchangewith 119905119896 if and only if the relationship of the parametersis satisfied with

1205721198960 = 1205821198961205820 = (1205902119861)119896(1205902119861)0 (8)

Instead of the hypothesis that the diffusion parameter is aconstant and does not change with the stress the conclusionthat both the drift parameter 120582 and the diffusion parameter120590119861 depend on stress variables could be drawn based on theprevious derivation At the same time it was testified thata unit with high realization of the drift parameter wouldpossess a high degradation rate and a high variation in thedegradation path in theory which is in conformity with theviewpoint of [19] The relationship between parameters andaccelerated stress variables can be set up by accelerationmodels based on engineering background The frequentlyused acceleration model includes the Arrhenius model theinverse power model and the Eyring model whose expres-sions and acceleration factors are listed in Table 1 Specify120589(119878119896 | 120573) = exp(minus120573119878119896) 120589(119878119896 | 120573) = exp(minus120573 ln(119878119896)) and120589(119878119896 | 120573) = exp(minus120573119878119896)119878119896 to the three models separatelyThe accelerated model of drift parameter 120582 the diffusionparameter120590119861 and the accelerated factor1205721198960 can be uniformlywritten as

120582119896 = 120578120589119896 (9)

(1205902119861)119896 = 120581120589119896 (10)

1205721198960 = 1205891198961205890 (11)

where 120589119896 is the abbreviated form of 120589(119878119896 | 120573) for simplificationof the expressions

23 ADT with Random Effects The observed degradation forproducts from the same population may be very differentowing to unobservable factors [4 27] And there are threesources of variability contributing to the nondeterminism ofdegradation (1) temporal variability (2) unit-to-unite vari-ability and (3) measurement variability [28] The temporalvariability is referred to as the inherent stochastic character-istics of the standard Brownian motion The measurementvariability is usually consideredwhile the degradation param-eters were estimated The unit-to-unite variability is usuallymodeled as random effects of the degradation parameterIt is very difficult to model the accelerated processes if the

4 Mathematical Problems in Engineering

drift parameter 120582 and the diffusion parameter 120590119861 are bothconsidered as the randomparameters So like Peng and Tseng[29] Si et al [30 31] and Tsai et al [32] it is also assumedthat different units have different drift parameters while alldiffusion parameters have the same value under a certain

stress Then we have 120578 sim 119873(120583120578 1205902120578) where 120583120578 and 1205902120578 are themean and variance of the parameter 120578 separately

Considering the random effects and the effects of accel-erated stresses on the drift parameter 120582 and the diffusionparameter 120590119861 we have the PDF and CDF of FHT under stress119878119896 as

119891119878119896 (119905) = 119908radic21205871205963 (120581120589119896 + 12059021205781205892119896120596) exp(minus (119908 minus 120583120578120589119896120596)22120596 (120581120589119896 + 12059021205781205892119896120596)) 119889120596119889119905 (12)

119865119878119896 (119905) =Φ( 120583120578120589119896120596 minus 119908radic120596(120581120589119896 + 12059021205781205892119896120596)) + exp(2120583120578119908120581 + 2120590212057811990821205812 ) sdot Φ(minus21205902120578120589119896119908120596 + 120581 (120583120578120589119896120596 + 119908)120581radic120581120589119896120596 + 120590212057812058921198961205962 ) if 119905 lt infinΦ(120583120578120590120578) + exp(2120583120578119908120581 + 2120590212057811990821205812 )Φ(minus21205902120578119908 + 120581120583120578120581120590120578 ) if 119905 997888rarr infin

(13)

whereΦ(sdot) is the distribution function of the standard normaldistribution

When 120583120578120590120578 approach infinity the transformed FHTunder stress 119878119896 is subject to inverse Gaussian (IG) distribu-tion

3 Model the CSADP and SSADP withRandom Effects

As themost-usedADTCSADTand SSADThave beenwidelyresearched But the models are quite different while the driftparameter 120582 and the diffusion parameter 120590119861 all depend onstress variables especially for SSADT

31 Modeling the CSADP with Random Effects Let 1198781 119878119896 119878119870 denote119870 stress level higher than normal stresslevel 1198780 such that 1198781 lt sdot sdot sdot lt 119878119870 and the subscript 119896 iscorresponding to the 119896th stress level 119896 = 1 2 119870 Supposethat there are 119873119896 units of samples tested under a constantaccelerated stress 119878119896 and each sample is measured 119872119896119894 timesat the 119896th stress level 119894 = 1 2 119873119896 The degradationat transformed time 120596119896119894119895 where 120596119896119894119895 = Λ(119905119896119894119895) is 119909119896119894119895 119895 =1 2 119872119896119894 The degradation process can be formulated as

119909119896119894119895 (120596119896119894119895 | 119878119896) = 120578119894120589119896120596119896119894119895 + radic120581120589119896119861 (120596119896119894119895) (14)

32Modeling the SSADPwith RandomEffects Similarly withCSADT it was assumed that there are 119870 accelerated stresslevels 1198781 lt sdot sdot sdot 119878119896 lt sdot sdot sdot lt 119878119870 in the whole test But thenumber of the samples is only 119873 which is different fromsum119870119896=1119873119896 inCSADT that verifies the characteristic that SSADTneeds fewer samples compared with CSADT [33] Supposethat each sample is also measured 119872119896119894 times at the 119896thstress level 119894 = 1 2 119873 119896 = 1 2 119870 And 120591119896119894 is thetransformed time scale where 120591119896119894 = Λ(119905119896119894119872119896119894) at which thestress changes from the 119896th stress level to the (119896 + 1)th stresslevel of the 119894th sample 119896 = 1 2 119870 minus 1 Besides 1205911198940 equals

0 and 120591119894119870 is the transformed end time of the 119870-step stressaccelerated test of the 119894th sample

The degradation process for CSADT is the same asSSADT under accelerated stress 1198781

1199091119894119895 = 12057811989412058911205961119894119895 + radic1205811205891119861 (1205961119894119895) 0 le 1205961119894119895 le 1205911119894 (15)

Raise the accelerated stress up to 1198782 at transformed timepoint 1205911119894 for the 119894th product Then the degradation is drivenby the accelerated stress 1198782 and the corresponding parameterscan be obtained by (9) are (10) where the drift parameter andthe diffusion parameter are equal to 1205781198941205892 and 1205811205892 separatelyThus the degradation consisted of the degradation driven by1198781 and the degradation driven by 1198782

1199092119894119895 = 12057811989412058911205911198941 + radic1205811205891119861 (1205911198941) + 1205781198941205892 (1205961198942119895 minus 1205911198941)+ radic1205811205892119861 (1205961198942119895 minus 1205911198941) (16)

Similarly the accelerated stress is turning up to 1198783 attransformed timepoint 1205912119894 and the degradation can bewrittenas

1199093119894119895 = 2sum119899=1

[120578119894120589119899 (120591119894119899 minus 120591119894(119899minus1)) + radic120581120589119899119861 (120591119894119899 minus 120591119894(119899minus1))]+ 1205781198941205893 (1205961198943119895 minus 1205911198942) + radic1205811205893119861 (1205961198943119895 minus 1205911198942)

(17)

Mathematical Problems in Engineering 5

According to the analysis the degradation process ofSSADT can be formulated as

119909119896119894119895 =

1205781198941205891120596119894119896119895 + radic1205811205891119861 (120596119894119896119895) 1205911198940 le 120596119894119896119895 le 120591119894112057811989412058911205911198941 + radic1205811205891119861 (1205911198941) + 1205781198941205892 (120596119894119896119895 minus 1205911198941) + radic1205811205892119861 (120596119894119896119895 minus 1205911198941) 1205911198941 le 120596119894119896119895 le 1205911198942sdot sdot sdot119896minus1sum119899=1

[120578119894120589119899 (120591119894119899 minus 120591119894(119899minus1)) + radic120581120589119899119861 (120591119894119899 minus 120591119894(119899minus1))] + 120578119894120589119896 (120596119894119896119895 minus 1205911198942) + radic120581120589119896119861 (120596119894119896119895 minus 1205911198942) 120591119894119896minus1 le 120596119894119896119895 le 120591119894119896sdot sdot sdot119870minus1sum119899=1

[120578119894120589119899 (120591119894119899 minus 120591119894(119899minus1)) + 120581120589119899119861 (120591119894119899 minus 120591119894(119899minus1))] + 120578119894120589119870 (120596119894119896119895 minus 1205911198942) + 120581120589119870119861 (120596119894119896119895 minus 1205911198942) 120591119894119870minus1 le 120596119894119896119895 le 120591119894119870

(18)

4 Parameter Estimation

In real applications it is inevitable that some measurementerrors may be introduced during the observation process[34] When a measurement is taken the observed degrada-tion is shown as follows for both CSADT and SSADT

119910119896119894119895 (120596119896119894119895 | 119878119896) = 119909119896119894119895 (120596119896119894119895 | 119878119896) + 120576119896119894119895119896 = 1 119870 119894 = 1 119873119896 119895 = 1 119872119896119894 (19)

where the measurement errors 120576119896119894119895 are assumed to be iidrealizations of 120576119896119894119895 sim 119873(0 1205902120576 ) and mutually independent of119909119896119894119895 [7 34] and 119873119896 is equal to 119873 for all of the acceleratedstresses of SSADT

The unknown parameters in the models are Θ =120583120578 1205902120578 120581 120573 1205902120576 There are two methods to deal with theunknown parameters with each considering one of thetwo main properties of Brownian motion which could becharacterized as follows [35]

(P1) The increment Δ119909(120596) = 119909(120596 + Δ120596) minus 119909(120596) isindependent of the degradation 119909(120596) which meansthat if 0 le 1205961 le 1205962 le 1205963 le 1205964 then 119909(1205962) minus119909(1205961) and 119909(1205964) minus 119909(1205963) are independent randomvariables Meanwhile the increment Δ119909(120596) follows anormal distribution where the mean equals 120582Δ120596 andthe variance is 1205902119861Δ120596

(P2) According to (P1) the degradation119909(120596) follows a nor-mal distribution with the mean 120582120596 and the variance1205902119861120596

Even though our models concern the unit-to-unitevariability the essence of the Wiener process remains thesame Owing to space constraints this paper deals with theunknown parameter based on property (P2) in the case ofCSADT and property (P1) in the case of SSADT

41 Parameter Estimation of CSADT Specify 119879119896119894 = (1198791198961198941 119879119896119894119872119896119894)1015840 119879119896119894119895 = 120596119896119894119895 119884119896119894 = (1198841198961198941 119884119896119894119872119896119894)1015840 119884119896 =(1198841198961 119884119896119873119896) and 119884 = (1198841 119884119870) for 119896 = 1 119870 119894 =

1 119873119896 and 119895 = 1 119872119896119894 According to property (P1) andindependent assumption of Brownian motion 119884119896119894 follows amultivariate normal distribution with mean and variance

119896119894 = 120583120578120589119896119879119896119894Σ119896119894 = 1205902120578Σ119896119894 (20)

where

Σ119896119894 = Ω119896119894 + 1205892119896119879119896119894119879119879119896119894119876119896119894 =

[[[[[[[[

1205961198961198941 1205961198961198941 sdot sdot sdot 12059611989611989411205961198961198941 1205961198961198942 sdot sdot sdot 1205961198961198942 d1205961198961198941 1205961198961198942 sdot sdot sdot 120596119896119894119872119896119894

]]]]]]]]

Ω119896119894 = 120589119896119876119896119894 + 2120576119868119896119894

(21)

and 119868119896119894 is an identified matrix of order119872119896119894The log-likelihood function of unknown parameters Θ =120583120578 1205902120578 120573 2120576 is

ln 119871 (Θ | 119884)= minus12 ln (2120587) 119870sum

119896=1

119873119896sum119894=1

119872119894119896 minus 12 ln (1205902120578) 119870sum119896=1

119873119896sum119894=1

119872119896119894minus 12 119870sum119896=1

119873119896sum119894=1

10038161003816100381610038161003816Σ11989611989410038161003816100381610038161003816minus 121205902120578

119870sum119896=1

119873119896sum119894=1

(119910119896119894 minus 120583120578120589119896119879119896119894)1015840 Σminus1119896119894 (119910119896119894 minus 120583120578120589119896119879119896119894)

(22)

6 Mathematical Problems in Engineering

Taking the first partial derivatives of the log-likelihoodfunction with respect to 120583120578 1205902120578 yields

120597 ln 119871 (Θ | 119884)120597120583120578= 11205902120578 (

119870sum119896=1

119873119896sum119894=1

1205891198961198791015840119896119894Σminus1119896119894 119910119896119894 minus 120583120578 119870sum119896=1

119873119896sum119894=1

12058921198961198791015840119896119894Σminus1119896119894 119879119896119894) 120597 ln 119871 (Θ | 119884)1205971205902120578 = minus 121205902120578

119870sum119896=1

119873119896sum119894=1

119872119896119894 + 12 (1205902120578)2sdot 119870sum119896=1

119873119896sum119894=1

(119910119896119894 minus 120583120578120589119896119879119896119894)1015840 Σminus1119896119894 (119910119896119894 minus 120583120578120589119896119879119896119894)

(23)

For the special value of ( 120573 2120576) setting the derivation of120597 ln 119871(Θ | 119884)120597120583120578 120597 ln 119871(Θ | 119884)1205971205902120578 to zero the MLE for120583120578 1205902120578 can be expressed as

120578 = sum119870119896=1sum119873119896119894=1 1205891198961198791015840119896119894Σminus1119896119894 119910119896119894sum119870119896=1sum119873119896119894=1 12058921198961198791015840119896119894Σminus1119896119894 119879119896119894 2120578 = 1sum119870119896=1sum119873119896119894=1119872119896119894

sdot 119870sum119896=1

119873119896sum119894=1

(119910119896119894 minus 120578120589119896119879119896119894)1015840 Σminus1119896119894 (119910119896119894 minus 120578120589119896119879119896119894) (24)

Substituting (24) into (22) and simplifying the profilelog-likelihood function can be written as

ln 119871 (Θ | 119884) = minus12 (ln (2120587) + 1) 119870sum119896=1

119873119896sum119894=1

119872119894119896minus 12 ln (2120578) 119870sum

119896=1

119873119896sum119894=1

119872119896119894minus 12 119870sum119896=1

119873119896sum119894=1

10038161003816100381610038161003816Σ11989611989410038161003816100381610038161003816 (25)

The MLE of 120573 and 2120576 can be obtained by maximizingthe profile log-likelihood function in (25) through a three-dimensional search Then substitute them into (24) we canobtain MLE for 120583120578 and 1205902120578

The value of and 2120576 can be obtained by the followingequations

= 120581 sdot 21205782120576 = 1205902120576 sdot 2120578 (26)

42 Parameter Estimation of SSADT The degradation pro-cess of SSADT shown as (18) is relatively complicated Thuswe introduce the accelerated factor first for the sake ofsimplicity The variable 120577119896 is represented by the product of 1205771and the accelerated factor 1205721198961

120589119896 = 12058911205721198961 (27)

The SSADP can be rewritten as

119909119896119894119895

=

12057811989412058911205961119894119895 + radic1205811205891119861 (12059612119895) 1205911198940 le 120596119894119896119895 le 12059111989411205781198941205891 (1205911119894 + 12057221 (1205962119894119895 minus 1205911119894)) + radic1205811205891119861 (1205911119894 + 12057221 (1205962119894119895 minus 1205911119894)) 1205911119894 le 1205962119894119895 le 1205912119894sdot sdot sdot1205781198941205891(119896minus1sum

119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119896119894119895 minus 120591(119896minus1)119894)) + radic1205811205891119861(119896minus1sum119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119896119894119895 minus 120591(119896minus1)119894)) 120591(119896minus1)119894 le 120596119896119894119895 le 120591119896119894sdot sdot sdot1205781198941205891(119870minus1sum

119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119870119894119895 minus 120591(119870minus1)119894)) + radic1205811205891119861(119870minus1sum119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119870119894119895 minus 120591(119870minus1)119894)) 120591(119870minus1)119894 le 120596119870119894119895 le 120591119870119894

(28)

Specify 120594119894119896119895 = sum119896minus1119899=1 1205721198991(120591119894119899 minus 120591119894(119899minus1)) + 12057221(120596119894119896119895 minus 120591119894(119896minus1))then the SSADP can be expressed a general formula asfollows

119909119896119894119895 = 1205781198941205891120594119896119894119895 + radic1205811205891119861 (120594119896119894119895) 120591(119896minus1)119894 le 120596119896119894119895 ge 120596119896119894119896 = 1 119870 119894 = 1 119873 119895 = 1 119872119896119894(29)

The foregoing transformation is equivalent to convertingthe degradation driven by stress 119878119896 to the degradation understress 1198781 in physics and the converted equivalent time is 120594119896119894119895At the moment the matrix of degradation and equivalenttransformed time are still bidimensional Because of thecontinuity of the degradation process the matrix can bewritten as a column vector with119872119894 element where119872119894 equalsthe sum of119872119894119896 119896 = 1 119870 and the subscript can be written

Mathematical Problems in Engineering 7

as 119898 for the 119894th product 119898 = 1 119872119894 Then the observeddegradation can be expressed as

119910119894119898 = 119909119894119898 + 120576119894119898119909119894119898 = 1205781198941205891120594119894119898 + 1205811205891119861 (120594119894119898) 1 le 119898 le 119870sum

119896=1

119872119894119896(30)

Define Δ1205941198941 = 1205941198941 Δ120594119894119898 = 120594119894119898 minus 120594119894(119898minus1) Δ120594119894 =(Δ1205941198941 Δ120594119894119872119894)1015840 119877119894 = Δ120594119894 and Δ119910119894119898 = 119910119894119898 minus 119910119894(119898minus1) for119894 = 1 119873 119898 = 1 119872119894 Then Δ119910119894 = (Δ1199101198941 Δ119910119894119872119894)1015840follows a multivariate normal distribution 119873(1205831205781205891119877119894 Π119894)where

Π119894 = 119882119894 + 1205902120578120589211198771198941198771015840119894 (119882119901119902)119894 = cov (Δ119910119894119901 Δ119910119894119902)

=

1205811205891119877119894119901 + 1205902120576 119901 = 119902 = 11205811205891119877119894119901 + 21205902120576 119901 = 119902 gt 1minus1205902120576 119901 = 119902 + 1 or 119901 = 119902 minus 10 otherwise

(31)

So the log-likelihood function can be expressed as

ln 119871 (Θ | 119884)= minus12 ln (2120587) 119873sum

119894=1

119872119894 minus 12 119873sum119894=1

ln (1003816100381610038161003816Π1198941003816100381610038161003816)minus 12 119873sum119894=1

(Δ119910119894 minus 1205831205781205891119877119894)1015840Πminus1119894 (Δ119910119894 minus 1205831205781205891119877119894) (32)

where1003816100381610038161003816Π1198941003816100381610038161003816 = 10038161003816100381610038161198821198941003816100381610038161003816 (1 + 1205902120578120589211198771015840119894119882minus1119894 119877119894)Πminus1119894 = 119882minus1119894 minus 1205902120578120589211 + 1205902120578120589211198771015840119894119882minus1119894 119877119894119882minus1119894 1198771198941198771015840119894119882minus1119894 (33)

Taking the first partial derivatives of the log-likelihoodfunction with respect to 120583120578 1205902120578 yields

120597 ln 119871 (Θ | 119884)120597120583120578 = 1205891 119873sum119894=1

1198771015840119894Πminus1119894 Δ119910119894 minus 12058312057812058921 119873sum119894=1

1198771015840119894Πminus1119894 119877119894 (34)

120597 ln 119871 (Θ | 119884)1205971205902120578 = minus 119873sum119894=1

120589211198771015840119894119882minus1119894 1198771 + 1205902120578120589211198771015840119894119882minus1119894 119877119894+ 119873sum119894=1

12058921 (Δ119910119894 minus 1205831205781205891119877119894)1015840119882minus1119894 1198771198941198771015840119894119882minus1119894 (Δ119910119894 minus 1205831205781205891119877119894)(1 + 1205902120578120589211198771015840119894119882minus1119894 119877)2(35)

For the special value of (120581 120573 1205902120576 ) setting the derivation of120597 ln 119871(Θ | 119884)120597120583120578 to zero the restricted MLE for 120583120578 can beexpressed as

120578 = sum119873119894=1 1198771015840119894Πminus1119894 Δ1199101198941205891sum119873119894=1 1198771015840119894Πminus1119894 119877119894 (36)

Similarly set the derivation of 120597 ln 119871(Θ | 119884)120597120583120578 to zeroand the results of the MLE for 1205902120578 will be discussed on twocases

Case 1 It was assumed that the number of the measurementsand the measurement points of each sample are the same forall of the samples under all of the accelerated stressThat is tosay the subscript of 119877119894 Π119894 and119882119894 can be removed

Thus the restricted MLE for 120583120578 can be expressed as

120578 = sum119873119894=1 1198771015840Πminus1119894 Δ11991011989412058911198731198771015840Πminus1119894 119877 (37)

The first partial derivatives of the log-likelihood functionto 1205902120578 can be rewritten as

120597 ln 119871 (Θ | 119884)1205971205902120578 = minus119873 120589211198771015840119882minus11198771 + 1205902120578120589211198771015840119882minus1119877+ 12058921(1 + 1205902120578120589211198771015840119882minus1119877)2sdot 119873sum119894=1

(Δ119910119894 minus 1205831205781205891119877)119882minus11198771198771015840119882minus1 (Δ119910119894 minus 1205831205781205891119877)1015840 (38)

For the special value of (120581 120573 1205902120576 ) setting the derivation of120597 ln 119871(Θ | 119884)1205971205902120578 to zero the result of the MLE for 1205902120578 can beexpressed as2120578

= sum119873119894=1 (Δ119910119894 minus 1205831205781205891119877)119882minus11198771198771015840119882minus1 (Δ119910119894 minus 1205831205781205891119877)101584011987312058921 (1198771015840119882minus1119877)2minus 1120589211198771015840119882minus1119877

(39)

Substituting (37) (39) into (32) and simplifying the profilelog-likelihood function can be written as

ln 119871 (120581 120573 1205902120576 | 120578 2120578 119884) = minus1198721198732 ln (2120587) minus 1198732 minus 1198732sdot ln (|119882|)minus 12

119873sum119894=1

Δ1199101015840119894119882minus1Δ119910119894 minus sum119873119894=1 (1198771015840119882minus1Δ119910119894)21198771015840119882minus1119877 minus 12sdot ln

sum119873119894=1 (1198771015840119882minus1Δ119910119894)21198731198771015840119882minus1119877 minus sum119873119894=1 (1198771015840119882minus1Δ119910119894)211987321198771015840119882minus1119877

(40)

8 Mathematical Problems in Engineering

where119872 is themeasurement time point of each samplewhichis the same for all of the samples based on the assumption ofCase 1

The MLE of 120581 120573 and 1205902120576 can be obtained by maximizingthe profile log-likelihood function in (40) through a three-dimensional searchThen substitute them into (37) and (39)we can obtain the MLE for 120583120578 and 1205902120578 Case 2 The number of the measurements and the measure-ment points of each sample are different for all of the samplesunder all of the accelerated stress In this case the first partialderivatives of the log-likelihood function to 1205902120578 are shown asin (35)There may be no analytical form by setting the partialderivatives to zero The evaluation of 120583120578 can be expressed as(36) Then the profile log-likelihood function of 1205902120578 120581 120573 1205902120576can be written as

ln 119871 (120581 120573 120590120576 1205902120578 | 120578 119884)= minus12 ln (2120587) 119873sum

119894=1

119872119894 minus 12 119873sum119894=1

ln (1003816100381610038161003816Π1198941003816100381610038161003816)minus 119873sum119894=1

(Δ119910119894 minus 1205781205891119877119894)Πminus1119894 (Δ119910119894 minus 1205781205891119877119894)1015840 (41)

The MLE of 1205902120578 120581 120573 and 1205902120576 can be obtained by maxi-mizing the profile log-likelihood function in (41) through afour-dimensional search Then substitute them into (36) wecan obtain the MLE for 120583120578

It is not to say that we can only use degradation forCSADT and increment for SSADT but just make an intro-duction to both of the two methods in the limited space Inaddition we could verify the results by comparing the esti-mation calculated by the two methods to avoid computationerrors

5 Simulation Data Analysis

In order to validate the model described before and theparameter estimation methods simulation test was con-ducted

51 The Simulation Method of CSADT Data The parametersΘ = 120583120578 1205902120578 120581 120573 1205902120576 should be given before the simulationand simulation process is shown as in the following steps

Step 1 Set 119896 = 1Step 2 Generate 119873119894 random numbers with subscripts 119894 119894 =1 2 119873119896 subject to normal distribution with mean 120583120578 andvariance 1205902120578 Step 3 Let 119897 = 0Step 4 Calculate119883119896119894(119897+1) using the Euler approximation [36]

119883119896119894(119897+1) = 119883119896119894119897 + 120578119894120589119896120596 (Δ119905) + 120581120589119896radic120596 (Δ119905)Ψ (42)

where Ψ is the random number according to the standardnormal distribution and Δ119905 is the step size in simulation and1198831198961198940 = 0 As such set 119897 = 119897 + 1 and return to Step 4 to get thedegradation path until 119897 = 119871119896119894+1 where 119871119896119894 is the simulationsteps for 119894th under stress 119878119896Step 5 Set 119896 = 119896 + 1 and return to Step 2 until 119896 = 119870 + 1Step 6 Extract the degradation based on the predefinedmeasurement time point 119905119896119894119895 where 119896 = 1 2 119870 119894 =1 2 119873119896 and 119895 = 1 2 119872119896119894 from the data set 119883119896119894119897 andget the data set of degradation119883119896119894119895Step 7 Simulate the measured degradation 119884119896119894119895 = 119883119896119894119895 + 120576119896119894119895where 120576119896119894119895 sim 119873(0 1205902120576 )52 The Simulation Method of SSADT Data There are somedifferences in the simulation process for SSADT comparedwith CSADT The simulation process is shown as follows

Step 1 Generate 119873 random numbers with subscripts 119894 119894 =1 2 119873 subject to normal distribution with mean 120583120578 andvariance 1205902120578 Step 2 Let 119896 = 1Step 3 Set 119897 = 1Step 4 Calculate119883119896119894(119897+1)

119883119896119894(119897+1) = 119883119896119894119897 + 120578119894120589119896120596 (Δ119905) + 120581120589119896radic120596 (Δ119905)Ψ (43)

where Ψ is the random number according to the standardnormal distribution and Δ119905 is the step size in simulation and1198831198961198940 = 01198831198961198940 = 119883(119896minus1)119894119871(119896minus1) (119896 gt 1) As such set 119897 = 119897+1 andreturn to Step 4 to get the degradation path until 119897 = 119871119896119894 + 1Step 5 Set 119896 = 119896 + 1 and return to Step 2 until 119896 = 119870 + 1Step 6 Extract the degradation based on the predefinedmeasurement time point 119905119896119894119895 where 119896 = 1 2 119870 119894 =1 2 119873 and 119895 = 1 2 119872119894119896 from the data set 119883119896119894119897 andget the data set of degradation119883119896119894119895Step 7 Simulate the measured degradation 119884119896119894119895 = 119883119896119894119895 + 120576119896119894119895where 120576119896119894119895 sim 119873(0 1205902120576 )53 The Analysis of the Simulated SSADT Data We justgive the analysis of simulated SSADT data here becausewe would give a case study of the CSADT later It wasassumed that the accelerated stress is temperature and thesimulation test contains 4 stresseswhich are 50∘C 60∘C 70∘Cand 80∘C For simplicity the transformed time function isset as Λ(119905) = 119905 and 20 degradation paths are generatedThen the parameters for degradation process are assumedas Θ = 5 4 025 minus3000 001 Moreover the time intervalmeasurement is 25 h for each sample of the whole simulationtest with 40 measurement time points The degradation dataare depicted as in Figure 1

Mathematical Problems in Engineering 9

Table 2 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 log-LF AIC RETruth value 5 1 025 mdash minus3000 001 mdash mdash mdash1198720 501 139 028 mdash minus297561 00078 142228 minus283456 084021198721 512 175 mdash 12907 lowast 10minus4 minus298081 00129 112289 minus223577 562711198722 620 mdash mdash 15213 lowast 10minus4 minus305175 13789 lowast 10minus9 109361 minus217721 78049

0 200 400 600 800 1000

0

05

1

15

2

25

th

y

Figure 1 The simulation degradation paths of SSADT

For simplicity the degradation model for SSADT pro-posed in this paper is referred to as1198720 the model presentedby Tang et al [17] as 1198721 and the stochastic SSADT modeldescribed in reference [13] as 1198722 The estimation results ofthe unknown parameters the log-likelihood function valueand the Akaike information criterion (AIC) [37] which isevaluated by (44) are summarized in Table 2 As we cansee from Table 2 our model clearly outperforms model interms of the log-LF and AIC The estimates of 120583120578 1205902120578 and1205902120576 are more close to the truth values compared with 1198721and 1198722 Meanwhile the diffusion-related variables includeparameters 120581 and 120573 and the diffusion parameter under thefour stresses can be obtained by (10) which is (28377 lowast 10minus548534 lowast 10minus5 78244 lowast 10minus5 12001 lowast 10minus4) in 1198721 and(22334lowast10minus5 39912lowast10minus5 64597lowast10minus5 99423lowast10minus5) forthe true model Moreover taking 1198781 as a reference a criterionformulizing the relative error (RE) is shown as (45) It is clearthat1198720 has the smallest RE from Table 2

AIC (119873) = minus2 ln 119871 (Θ) + 2119873 (44)

RE = 1003816100381610038161003816100381610038161003816100381610038161003816120583120578 minus 120578120578

1003816100381610038161003816100381610038161003816100381610038161003816 +100381610038161003816100381610038161003816100381610038161003816100381610038161205902120578 minus 21205782120578

10038161003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161003816(1205902119861)1 minus (2119861)1(2119861)1

10038161003816100381610038161003816100381610038161003816100381610038161003816+ 1003816100381610038161003816100381610038161003816100381610038161003816120573 minus

1003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161205902120576 minus 21205762120576

1003816100381610038161003816100381610038161003816100381610038161003816 (45)

It is assumed that the failure threshold119908 is equal to 5 andthe normal stress is 25∘C The PDF and CDF under normalstress are shown as in Figure 2 The phenomenon can beobserved where the PDF and CDF of our model most nearlyapproached the real model By contrasting 1198721 with 1198722 itcan be concluded that the neglect of random effects wouldresult in sharper PDF which is consistent with the result inreference [17] And it is noteworthy that the 05 quantiles ofthe failure life are almost equal from the CDF of the threemodels But there is a greater difference between the realmodel and 1198721 while the indicator is taken as mean time tofailure as well as the difference between the real model and1198722 So the correlation between the stress and the diffusionparameter should not be neglected Next we are going toprove the superiority of our model in practical application

6 Case Application

The CSADT model with covariates and random effects isverified by the accelerated degradation data of carbon-filmresistors whose raw data set is explicitly given in Table C3of Meeker and Escobar [38]There were 29 samples and threeaccelerated temperature stresses in the whole test where ninesamples were observed at 83∘C the remaining two stresses133∘C and 173∘C had 10 samples for each stress All of thesamples were observed at the same time points with 1199050 = 01199051 = 452 1199052 = 1030 1199053 = 4341 and 1199054 = 8084 (inhours) Similar to [39] it was assumed that the standardoperating temperature was 50∘C and the threshold value forpercent increase in resistance was taken to be 119862 = 12 Itis reasonable and computationally easier to use the ratios inresistance to the initial value for each rather than the percentincrease [9] For ease of calculation the logarithm of the ratiowas taken as the degradation here That is the value 028of the percent increase was changed to 2796119864 minus 3 and soon and the transformed threshold value 119908 was 01133 Thedegradation paths of all the sample were as shown in Figure 3It can be seen that the degradation of the samples uniformlyshowed a nonlinear characteristic especially at the beginningof the ADT Thus the degradation is modeled with the timescale transformedWiener process as (1) with 120596 = Λ(119905) = 119905119887

Similarly the proposed degradation model for SSADT inthis paper is referred to as1198720 themodel presented by Tang etal [17] as1198721 and themodel described in reference [9] as1198723The estimation results of the unknown parameters the log-likelihood function value and the AIC are shown in Table 3The estimated parameters 119887 are approximately equal for thethree models Take 119887 = 050 as an example to show the time

10 Mathematical Problems in Engineering

Table 3 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 b Log-LF AIC MTTF1198720 823 207 108119864 minus 02 mdash minus420282 182119864 minus 20 050 51848 minus102497 380119864 + 071198721 1471 433 mdash 413119864 minus 07 minus458656 506119864 minus 20 053 48637 minus96073 515119864 + 071198722 1111 mdash mdash 676119864 minus 07 minus447982 411119864 minus 18 053 46837 minus92674 406119864 + 07

0 1 2 3 4 50

05

1

15

2

25

th

The P

DF

Real model

times104

times10minus4

M1

M0

M2

0 1 2 3 4 50

02

04

06

08

1

th

The C

DF

times104

Real modelM1

M0

M2

Figure 2 The comparison of the PDF and CDF of the three models for simulated SSADT data

0 2000 4000 6000 80000

0002

0004

0006

0008

001

th

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 2000 4000 6000 80000

001

002

003

th

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 2000 4000 6000 8000th

0

002

004

006

008

01

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 3 The degradation paths of carbon-film resistors

scale transformed degradation paths as in Figure 4 which areapproximately straight lines compared with Figure 3

Our model has the largest log-LF and smallest AICcompared with1198721 and1198722 The result proves that our modelis more suitable for the degradation data Besides 1198721 hasthe larger log-LF and smaller AIC compared with1198722 So therandom effects could not be neglected It can be observed thatthe variance of the drift parameter is slightly largerwhichmaybe owing to the fixed value of diffusion parameter in1198721

The PDF and CDF under the standard operating temper-ature are as shown in Figure 5The time corresponding to thepeak values of PDF were as follows 2676 lowast 107 1049 lowast 107

and 5028lowast106 for1198720 to1198722 But theMTTF of1198720 isminimalfrom Table 3 And the uncertainty in the estimated PDFs ofthe lifetimes under1198720 is smallest compared to1198721 and1198722 asseen in Figure 5When time 119905 approaches positive infinity theCDF of ourmodel which is calculated by (13) tends illimitablyto 1 and the CDF of 1198721 is roughly equal to 09997 It followsthat our model could do well in distinguishing the differencefrom random effects and the effect of covariates Anotherthing which is worth noting is that the measurement error isquite tiny in the case but it is just an individual phenomenonThemeasurement error should also be considered in practicalapplication

Mathematical Problems in Engineering 11

0 20 40 60 800

0002

0004

0006

0008

001

Λ(t)

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 20 40 60 800

001

002

003

Λ(t)

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 20 40 60 800

002

004

006

008

01

Λ(t)

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 4 The degradation paths of carbon-film resistors under transformed time scale

0 2 4 6 8 100

05

1

15

2

25

3

35

th

times10minus8

M0

M1

M2

The P

DF

times107

0 2 4 6 8 100

02

04

06

08

1

th

M0

M1

M2

The C

DF

times107

Figure 5 The comparison of the PDF and CDF of the three models for real CSADT data

7 Conclusion

In this paper the degradation models based on nonlinearWiener process are established for both constant stress accel-erated degradation data and step stress accelerated degrada-tion data Before the establishment the relationship betweenthe drift parameter and stress variables is derived based onthe invariance principle of failure mechanism and Nelsonassumption so is the relationship between the diffusionparameter and stress variables It is concluded that the ratioof drift parameters under two stresses is a constant which isirrelevant to the testing time and depends only on the twostresses as long as the ratio of diffusion parameters is equalto the ratio of drift parameters And the ratio is defined asaccelerated factor Besides the random effects are also takeninto consideration where the drift parameter is assumed to be

normally distributed and the diffusion parameter is same forall of the samples under a certain stress Then the PDF andCDF of the FHT are deduced considering random effects

Because of the dependency between the diffusion param-eter and stress variables the degradation process is quitedifferent either for CSADT or for SSADT The CSADP andSSADP with random effects are modeled Moreover theunknown parameters are solved by MLE based on the twoproperties of Wiener process At the end of the paper thesimulated data of SSADT and the CSADTdata of carbon-filmresistors are both analyzed to verify the proposed model Itis concluded that the model has the biggest log-LF and thesmallest AIC compared with the two other models

The innovation of this paper lies in the following Firstthe random effects are considered under the new relationshipbetween the diffusion parameter and accelerated stresses

12 Mathematical Problems in Engineering

Second the degradation process was modeled for bothCSADT and SSADT Thirdly the unknown parameters wereestimated based on the two properties of Wiener process andthe result of theMLE for 1205902120578 is discussed on two cases Fourththe measurement error of the degradation data is also valued

However we have only considered the random effects ofthe drift parameter in this paper due to the complexity of thecomputation A further research may consider the randomeffects of the diffusion parameter into the model At the sametime the study of the paper may provide new ideas for therelativity analysis between the parameters of other stochasticprocess and stress variables

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by Jiangsu Province GraduateStudent Scientific Research Innovation Project of China(Project KYLX15 0330)The help is gratefully acknowledged

References

[1] H-F Yu ldquoOptimal classification of highly-reliable productswhose degradation paths satisfyWiener processesrdquo EngineeringOptimization vol 35 no 3 pp 313ndash324 2003

[2] XWang N Balakrishnan and B Guo ldquoResidual life estimationbased on a generalized Wiener degradation processrdquo ReliabilityEngineering and System Safety vol 124 pp 13ndash23 2014

[3] Y Ren Q Feng T Ye and B Sun ldquoA novel model of reliabilityassessment for circular electrical connectorsrdquo IEEE Transac-tions on Components Packaging andManufacturing Technologyvol 5 no 6 pp 755ndash761 2015

[4] Z-S Ye and M Xie ldquoStochastic modelling and analysis ofdegradation for highly reliable productsrdquo Applied StochasticModels in Business and Industry vol 31 no 1 pp 16ndash32 2015

[5] H Lim and B-J Yum ldquoOptimal design of accelerated degrada-tion tests based on Wiener process modelsrdquo Journal of AppliedStatistics vol 38 no 2 pp 309ndash325 2011

[6] C J Lu and W Q Meeker ldquoUsing degradation measures toestimate a time-to-failure distributionrdquo Technometrics vol 35no 2 pp 161ndash174 1993

[7] G AWhitmore ldquoEstimating degradation by a wiener diffusionprocess subject to measurement errorrdquo Lifetime Data Analysisvol 1 no 3 pp 307ndash319 1995

[8] Z Pan and N Balakrishnan ldquoReliability modeling of degra-dation of products with multiple performance characteristicsbased on gamma processesrdquo Reliability Engineering amp SystemSafety vol 96 no 8 pp 949ndash957 2011

[9] C Park and W J Padgett ldquoAccelerated degradation modelsfor failure based on geometric Brownian motion and gammaprocessesrdquo Lifetime Data Analysis vol 11 no 4 pp 511ndash5272005

[10] Z-S Ye and N Chen ldquoThe inverse Gaussian process as adegradation modelrdquo Technometrics vol 56 no 3 pp 302ndash3112014

[11] A Onar and W J Padgett ldquoAccelerated test models with theinverse Gaussian distributionrdquo Journal of Statistical Planningand Inference vol 89 no 1-2 pp 119ndash133 2000

[12] T-R Tsai Y L Lio and N Jiang ldquoOptimal decisions on theaccelerated degradation test plan under the Wiener processrdquoQuality Technology and Quantitative Management vol 11 no4 pp 461ndash470 2014

[13] C-M Liao and S-T Tseng ldquoOptimal design for step-stressaccelerated degradation testsrdquo IEEE Transactions on Reliabilityvol 55 no 1 pp 59ndash66 2006

[14] L C Tang G Yang and M Xie ldquoPlanning of step-stress accel-erated degradation testrdquo in Proceedings of the Annual Reliabil-ity and Maintainability SymposiummdashProceedings InternationalSymposium on Product Quality and Integrity pp 287ndash292January 2004

[15] C-H Hu M-Y Lee and J Tang ldquoOptimum step-stress accel-erated degradation test for Wiener degradation process underconstraintsrdquo European Journal of Operational Research vol 241no 2 pp 412ndash421 2015

[16] S Tang ldquoStep stress accelerated degradation process modelingand remaining useful life estimationrdquo Journal of MechanicalEngineering vol 50 no 16 p 33 2014

[17] S Tang X Guo C Yu H Xue and Z Zhou ldquoAccelerated degra-dation tests modeling based on the nonlinear wiener processwith random effectsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 560726 11 pages 2014

[18] L Wang R Pan X Li and T Jiang ldquoA Bayesian reliability eval-uation method with integrated accelerated degradation testingand field informationrdquo Reliability Engineering amp System Safetyvol 112 pp 38ndash47 2013

[19] Z-S Ye N Chen and Y Shen ldquoA new class of Wiener processmodels for degradation analysisrdquo Reliability Engineering ampSystem Safety vol 139 pp 58ndash67 2015

[20] G A Whitmore and F Schenkelberg ldquoModelling accelerateddegradation data using wiener diffusion with a time scaletransformationrdquo Lifetime Data Analysis vol 3 no 1 pp 27ndash451997

[21] K A Doksum and A Hoyland ldquoModels for variable-stressaccelerated life testing experiments based on Wiener processesand the inverse Gaussian distributionrdquo Technometrics vol 34no 1 pp 74ndash82 1992

[22] H Liao and E A Elsayed ldquoReliability inference for field con-ditions from accelerated degradation testingrdquo Naval ResearchLogistics vol 53 no 6 pp 576ndash587 2006

[23] H-WWang T-X Xu andW-YWang ldquoRemaining life predic-tion based on wiener processes with ADT prior informationrdquoQuality and Reliability Engineering International vol 32 no 3pp 753ndash765 2015

[24] C-Y Peng ldquoInverse Gaussian processes with random effectsand explanatory variables for degradation datardquo Technometricsvol 57 no 1 pp 100ndash111 2015

[25] X Pan XHuang Y Chen YWang and R Kang ldquoConnotationof failure mechanism consistency and identificationmethod foraccelerated testingrdquo in Proceedings of the Prognostics and SystemHealth Management Conference (PHM-Shenzhen rsquo11) May 2011

[26] W B Nelson Accelerated Testing Statistical Models Test Plansand Data Analysis John Wiley amp Sons 2009

[27] Z-S Ye Y Hong and Y Xie ldquoHow do heterogeneities in oper-ating environments affect field failure predictions and testplanningrdquo The Annals of Applied Statistics vol 7 no 4 pp2249ndash2271 2013

Mathematical Problems in Engineering 13

[28] Z-X Zhang X-S Si C-H Hu Q Zhang T Li and C XuldquoPlanning repeated degradation testing for products with three-source variabilityrdquo IEEE Transactions on Reliability vol 65 no2 pp 640ndash647 2016

[29] C-Y Peng and S-T Tseng ldquoMis-specification analysis of lineardegradation modelsrdquo IEEE Transactions on Reliability vol 58no 3 pp 444ndash455 2009

[30] X-S Si W Wang C-H Hu D-H Zhou and M G PechtldquoRemaining useful life estimation based on a nonlinear diffu-sion degradation processrdquo IEEE Transactions on Reliability vol61 no 1 pp 50ndash67 2012

[31] X-S Si W Wang C-H Hu M-Y Chen and D-H Zhou ldquoAWiener-process-based degradationmodel with a recursive filteralgorithm for remaining useful life estimationrdquo MechanicalSystems and Signal Processing vol 35 no 1-2 pp 219ndash237 2013

[32] C-C Tsai S-T Tseng and N Balakrishnan ldquoMis-specificationanalyses of gamma andWiener degradation processesrdquo Journalof Statistical Planning and Inference vol 141 no 12 pp 3725ndash3735 2011

[33] F Haghighi and S J Bae ldquoReliability estimation from lineardegradation and failure time data with competing risks under astep-stress accelerated degradation testrdquo IEEE Transactions onReliability vol 64 no 3 pp 960ndash971 2015

[34] Z-S Ye Y Wang K-L Tsui and M Pecht ldquoDegradation dataanalysis usingwiener processeswithmeasurement errorsrdquo IEEETransactions on Reliability vol 62 no 4 pp 772ndash780 2013

[35] J Huang D S Golubovic S Koh et al ldquoDegradation modelingof mid-power white-light LEDs by using Wiener processrdquoOptics Express vol 23 no 15 pp A966ndashA978 2015

[36] A Beskos O Papaspiliopoulos G O Roberts and P Fearn-head ldquoExact and computationally efficient likelihood-basedestimation for discretely observed diffusion processes (withdiscussion)rdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 68 no 3 pp 333ndash382 2006

[37] J Shang and J E Cavanaugh ldquoAn assumption for the develop-ment of bootstrap variants of the Akaike information criterionin mixed modelsrdquo Statistics and Probability Letters vol 78 no12 pp 1422ndash1429 2008

[38] W QMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[39] W J Padgett andM A Tomlinson ldquoInference from accelerateddegradation and failure data based on Gaussian process mod-elsrdquo Lifetime Data Analysis vol 10 no 2 pp 191ndash206 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Table 1 Three accelerated models and their acceleration factor

Accelerated models Drift parameter 120582 Diffusion parameter 120590119861 Acceleration factor

Arrhenius model 120582119896 = 120578 exp(minus 120573119878119896 ) (1205902119861)119896 = 120581 exp(minus 120573119878119896 ) 1205721198960 = exp(minus120573( 1119878119896 minus 11198780 ))Inverse power model 120582119896 = 120578 exp (minus120573 ln(119878119896)) (1205902119861)119896 = 120581 exp (minus120573 ln(119878119896)) 1205721198960 = exp (minus120573(ln(119878119896) minus ln(1198780)))Eyring model 120582119896 = 120578119878119896 exp (minus120573119878119896) (1205902119861)119896 = 120581119878119896 exp (minus120573119878119896) 1205721198960 = 1198780119878119896 exp (minus120573 (1119878119896 minus 11198780))degradation of products by elevating stress and the obtaineddegradation data are then used to extrapolate the informationthrough accelerating model to obtain the estimates of life orperformance of products at normal use condition To ensurethe accuracy of the extrapolation the failure mechanismunder the accelerated stress and the normal stress mustkeep the same which is also the premise of the ADT Oneof the most common methods for consistency inspectionof the failure mechanism is based on statistical method[25] The principle of this method is that the accelerationfactor is a constant and independent of testing time if thefailure mechanism remains unchanged The definition of theacceleration factor is given below according to the Nelsonassumption [26]

Specify 120596 = Λ(119905) and suppose 119865lowast represents thepredetermined cumulative failure probability 1205960 is definedas the testing time when the accumulated failure probabilitycomes to 119865lowast under normal stress 1198780 as well as 120596119896 underaccelerated stress 119878119896

119865lowast = 119865119896 (120596119896) = 1198650 (1205960) (4)

Then the acceleration factor 1205721198960 of stress 119878119896 relative tostress 1198780 can be defined as

1205721198960 = 1205960120596119896 (5)

The expression 1205960 = 1205721198960120596119896 can be obtained from (5)and plug it into (4) Then take the first-order derivative withrespect to 120596119896 and we have the following equation for any120596119896 gt 0

119891119896 (120596119896) = 12057211989601198910 (1205721198960120596119896) (6)

The expression of 119891119896(119905119896) and 1198910(1199050) can be deductedaccording to (2) then

1205721198960 = 119891119896 (120596119896)1198910 (1205721198960120596119896) = radic (1205721198960)3 (1205902119861)0(1205902119861)119896sdot exp[ 12120596119896 ( 1199082(1205902119861)0 1205721198960 minus 1199082(1205902119861)119896)+ 1205961198962 (120582201205721198960(1205902119861)0 minus 1205822119896(1205902119861)119896) + 119908( 120582119896(1205902119861)119896 minus 1205820(1205902119861)0)]

(7)

The acceleration factor 1205721198960 is a constant that does notchangewith 119905119896 if and only if the relationship of the parametersis satisfied with

1205721198960 = 1205821198961205820 = (1205902119861)119896(1205902119861)0 (8)

Instead of the hypothesis that the diffusion parameter is aconstant and does not change with the stress the conclusionthat both the drift parameter 120582 and the diffusion parameter120590119861 depend on stress variables could be drawn based on theprevious derivation At the same time it was testified thata unit with high realization of the drift parameter wouldpossess a high degradation rate and a high variation in thedegradation path in theory which is in conformity with theviewpoint of [19] The relationship between parameters andaccelerated stress variables can be set up by accelerationmodels based on engineering background The frequentlyused acceleration model includes the Arrhenius model theinverse power model and the Eyring model whose expres-sions and acceleration factors are listed in Table 1 Specify120589(119878119896 | 120573) = exp(minus120573119878119896) 120589(119878119896 | 120573) = exp(minus120573 ln(119878119896)) and120589(119878119896 | 120573) = exp(minus120573119878119896)119878119896 to the three models separatelyThe accelerated model of drift parameter 120582 the diffusionparameter120590119861 and the accelerated factor1205721198960 can be uniformlywritten as

120582119896 = 120578120589119896 (9)

(1205902119861)119896 = 120581120589119896 (10)

1205721198960 = 1205891198961205890 (11)

where 120589119896 is the abbreviated form of 120589(119878119896 | 120573) for simplificationof the expressions

23 ADT with Random Effects The observed degradation forproducts from the same population may be very differentowing to unobservable factors [4 27] And there are threesources of variability contributing to the nondeterminism ofdegradation (1) temporal variability (2) unit-to-unite vari-ability and (3) measurement variability [28] The temporalvariability is referred to as the inherent stochastic character-istics of the standard Brownian motion The measurementvariability is usually consideredwhile the degradation param-eters were estimated The unit-to-unite variability is usuallymodeled as random effects of the degradation parameterIt is very difficult to model the accelerated processes if the

4 Mathematical Problems in Engineering

drift parameter 120582 and the diffusion parameter 120590119861 are bothconsidered as the randomparameters So like Peng and Tseng[29] Si et al [30 31] and Tsai et al [32] it is also assumedthat different units have different drift parameters while alldiffusion parameters have the same value under a certain

stress Then we have 120578 sim 119873(120583120578 1205902120578) where 120583120578 and 1205902120578 are themean and variance of the parameter 120578 separately

Considering the random effects and the effects of accel-erated stresses on the drift parameter 120582 and the diffusionparameter 120590119861 we have the PDF and CDF of FHT under stress119878119896 as

119891119878119896 (119905) = 119908radic21205871205963 (120581120589119896 + 12059021205781205892119896120596) exp(minus (119908 minus 120583120578120589119896120596)22120596 (120581120589119896 + 12059021205781205892119896120596)) 119889120596119889119905 (12)

119865119878119896 (119905) =Φ( 120583120578120589119896120596 minus 119908radic120596(120581120589119896 + 12059021205781205892119896120596)) + exp(2120583120578119908120581 + 2120590212057811990821205812 ) sdot Φ(minus21205902120578120589119896119908120596 + 120581 (120583120578120589119896120596 + 119908)120581radic120581120589119896120596 + 120590212057812058921198961205962 ) if 119905 lt infinΦ(120583120578120590120578) + exp(2120583120578119908120581 + 2120590212057811990821205812 )Φ(minus21205902120578119908 + 120581120583120578120581120590120578 ) if 119905 997888rarr infin

(13)

whereΦ(sdot) is the distribution function of the standard normaldistribution

When 120583120578120590120578 approach infinity the transformed FHTunder stress 119878119896 is subject to inverse Gaussian (IG) distribu-tion

3 Model the CSADP and SSADP withRandom Effects

As themost-usedADTCSADTand SSADThave beenwidelyresearched But the models are quite different while the driftparameter 120582 and the diffusion parameter 120590119861 all depend onstress variables especially for SSADT

31 Modeling the CSADP with Random Effects Let 1198781 119878119896 119878119870 denote119870 stress level higher than normal stresslevel 1198780 such that 1198781 lt sdot sdot sdot lt 119878119870 and the subscript 119896 iscorresponding to the 119896th stress level 119896 = 1 2 119870 Supposethat there are 119873119896 units of samples tested under a constantaccelerated stress 119878119896 and each sample is measured 119872119896119894 timesat the 119896th stress level 119894 = 1 2 119873119896 The degradationat transformed time 120596119896119894119895 where 120596119896119894119895 = Λ(119905119896119894119895) is 119909119896119894119895 119895 =1 2 119872119896119894 The degradation process can be formulated as

119909119896119894119895 (120596119896119894119895 | 119878119896) = 120578119894120589119896120596119896119894119895 + radic120581120589119896119861 (120596119896119894119895) (14)

32Modeling the SSADPwith RandomEffects Similarly withCSADT it was assumed that there are 119870 accelerated stresslevels 1198781 lt sdot sdot sdot 119878119896 lt sdot sdot sdot lt 119878119870 in the whole test But thenumber of the samples is only 119873 which is different fromsum119870119896=1119873119896 inCSADT that verifies the characteristic that SSADTneeds fewer samples compared with CSADT [33] Supposethat each sample is also measured 119872119896119894 times at the 119896thstress level 119894 = 1 2 119873 119896 = 1 2 119870 And 120591119896119894 is thetransformed time scale where 120591119896119894 = Λ(119905119896119894119872119896119894) at which thestress changes from the 119896th stress level to the (119896 + 1)th stresslevel of the 119894th sample 119896 = 1 2 119870 minus 1 Besides 1205911198940 equals

0 and 120591119894119870 is the transformed end time of the 119870-step stressaccelerated test of the 119894th sample

The degradation process for CSADT is the same asSSADT under accelerated stress 1198781

1199091119894119895 = 12057811989412058911205961119894119895 + radic1205811205891119861 (1205961119894119895) 0 le 1205961119894119895 le 1205911119894 (15)

Raise the accelerated stress up to 1198782 at transformed timepoint 1205911119894 for the 119894th product Then the degradation is drivenby the accelerated stress 1198782 and the corresponding parameterscan be obtained by (9) are (10) where the drift parameter andthe diffusion parameter are equal to 1205781198941205892 and 1205811205892 separatelyThus the degradation consisted of the degradation driven by1198781 and the degradation driven by 1198782

1199092119894119895 = 12057811989412058911205911198941 + radic1205811205891119861 (1205911198941) + 1205781198941205892 (1205961198942119895 minus 1205911198941)+ radic1205811205892119861 (1205961198942119895 minus 1205911198941) (16)

Similarly the accelerated stress is turning up to 1198783 attransformed timepoint 1205912119894 and the degradation can bewrittenas

1199093119894119895 = 2sum119899=1

[120578119894120589119899 (120591119894119899 minus 120591119894(119899minus1)) + radic120581120589119899119861 (120591119894119899 minus 120591119894(119899minus1))]+ 1205781198941205893 (1205961198943119895 minus 1205911198942) + radic1205811205893119861 (1205961198943119895 minus 1205911198942)

(17)

Mathematical Problems in Engineering 5

According to the analysis the degradation process ofSSADT can be formulated as

119909119896119894119895 =

1205781198941205891120596119894119896119895 + radic1205811205891119861 (120596119894119896119895) 1205911198940 le 120596119894119896119895 le 120591119894112057811989412058911205911198941 + radic1205811205891119861 (1205911198941) + 1205781198941205892 (120596119894119896119895 minus 1205911198941) + radic1205811205892119861 (120596119894119896119895 minus 1205911198941) 1205911198941 le 120596119894119896119895 le 1205911198942sdot sdot sdot119896minus1sum119899=1

[120578119894120589119899 (120591119894119899 minus 120591119894(119899minus1)) + radic120581120589119899119861 (120591119894119899 minus 120591119894(119899minus1))] + 120578119894120589119896 (120596119894119896119895 minus 1205911198942) + radic120581120589119896119861 (120596119894119896119895 minus 1205911198942) 120591119894119896minus1 le 120596119894119896119895 le 120591119894119896sdot sdot sdot119870minus1sum119899=1

[120578119894120589119899 (120591119894119899 minus 120591119894(119899minus1)) + 120581120589119899119861 (120591119894119899 minus 120591119894(119899minus1))] + 120578119894120589119870 (120596119894119896119895 minus 1205911198942) + 120581120589119870119861 (120596119894119896119895 minus 1205911198942) 120591119894119870minus1 le 120596119894119896119895 le 120591119894119870

(18)

4 Parameter Estimation

In real applications it is inevitable that some measurementerrors may be introduced during the observation process[34] When a measurement is taken the observed degrada-tion is shown as follows for both CSADT and SSADT

119910119896119894119895 (120596119896119894119895 | 119878119896) = 119909119896119894119895 (120596119896119894119895 | 119878119896) + 120576119896119894119895119896 = 1 119870 119894 = 1 119873119896 119895 = 1 119872119896119894 (19)

where the measurement errors 120576119896119894119895 are assumed to be iidrealizations of 120576119896119894119895 sim 119873(0 1205902120576 ) and mutually independent of119909119896119894119895 [7 34] and 119873119896 is equal to 119873 for all of the acceleratedstresses of SSADT

The unknown parameters in the models are Θ =120583120578 1205902120578 120581 120573 1205902120576 There are two methods to deal with theunknown parameters with each considering one of thetwo main properties of Brownian motion which could becharacterized as follows [35]

(P1) The increment Δ119909(120596) = 119909(120596 + Δ120596) minus 119909(120596) isindependent of the degradation 119909(120596) which meansthat if 0 le 1205961 le 1205962 le 1205963 le 1205964 then 119909(1205962) minus119909(1205961) and 119909(1205964) minus 119909(1205963) are independent randomvariables Meanwhile the increment Δ119909(120596) follows anormal distribution where the mean equals 120582Δ120596 andthe variance is 1205902119861Δ120596

(P2) According to (P1) the degradation119909(120596) follows a nor-mal distribution with the mean 120582120596 and the variance1205902119861120596

Even though our models concern the unit-to-unitevariability the essence of the Wiener process remains thesame Owing to space constraints this paper deals with theunknown parameter based on property (P2) in the case ofCSADT and property (P1) in the case of SSADT

41 Parameter Estimation of CSADT Specify 119879119896119894 = (1198791198961198941 119879119896119894119872119896119894)1015840 119879119896119894119895 = 120596119896119894119895 119884119896119894 = (1198841198961198941 119884119896119894119872119896119894)1015840 119884119896 =(1198841198961 119884119896119873119896) and 119884 = (1198841 119884119870) for 119896 = 1 119870 119894 =

1 119873119896 and 119895 = 1 119872119896119894 According to property (P1) andindependent assumption of Brownian motion 119884119896119894 follows amultivariate normal distribution with mean and variance

119896119894 = 120583120578120589119896119879119896119894Σ119896119894 = 1205902120578Σ119896119894 (20)

where

Σ119896119894 = Ω119896119894 + 1205892119896119879119896119894119879119879119896119894119876119896119894 =

[[[[[[[[

1205961198961198941 1205961198961198941 sdot sdot sdot 12059611989611989411205961198961198941 1205961198961198942 sdot sdot sdot 1205961198961198942 d1205961198961198941 1205961198961198942 sdot sdot sdot 120596119896119894119872119896119894

]]]]]]]]

Ω119896119894 = 120589119896119876119896119894 + 2120576119868119896119894

(21)

and 119868119896119894 is an identified matrix of order119872119896119894The log-likelihood function of unknown parameters Θ =120583120578 1205902120578 120573 2120576 is

ln 119871 (Θ | 119884)= minus12 ln (2120587) 119870sum

119896=1

119873119896sum119894=1

119872119894119896 minus 12 ln (1205902120578) 119870sum119896=1

119873119896sum119894=1

119872119896119894minus 12 119870sum119896=1

119873119896sum119894=1

10038161003816100381610038161003816Σ11989611989410038161003816100381610038161003816minus 121205902120578

119870sum119896=1

119873119896sum119894=1

(119910119896119894 minus 120583120578120589119896119879119896119894)1015840 Σminus1119896119894 (119910119896119894 minus 120583120578120589119896119879119896119894)

(22)

6 Mathematical Problems in Engineering

Taking the first partial derivatives of the log-likelihoodfunction with respect to 120583120578 1205902120578 yields

120597 ln 119871 (Θ | 119884)120597120583120578= 11205902120578 (

119870sum119896=1

119873119896sum119894=1

1205891198961198791015840119896119894Σminus1119896119894 119910119896119894 minus 120583120578 119870sum119896=1

119873119896sum119894=1

12058921198961198791015840119896119894Σminus1119896119894 119879119896119894) 120597 ln 119871 (Θ | 119884)1205971205902120578 = minus 121205902120578

119870sum119896=1

119873119896sum119894=1

119872119896119894 + 12 (1205902120578)2sdot 119870sum119896=1

119873119896sum119894=1

(119910119896119894 minus 120583120578120589119896119879119896119894)1015840 Σminus1119896119894 (119910119896119894 minus 120583120578120589119896119879119896119894)

(23)

For the special value of ( 120573 2120576) setting the derivation of120597 ln 119871(Θ | 119884)120597120583120578 120597 ln 119871(Θ | 119884)1205971205902120578 to zero the MLE for120583120578 1205902120578 can be expressed as

120578 = sum119870119896=1sum119873119896119894=1 1205891198961198791015840119896119894Σminus1119896119894 119910119896119894sum119870119896=1sum119873119896119894=1 12058921198961198791015840119896119894Σminus1119896119894 119879119896119894 2120578 = 1sum119870119896=1sum119873119896119894=1119872119896119894

sdot 119870sum119896=1

119873119896sum119894=1

(119910119896119894 minus 120578120589119896119879119896119894)1015840 Σminus1119896119894 (119910119896119894 minus 120578120589119896119879119896119894) (24)

Substituting (24) into (22) and simplifying the profilelog-likelihood function can be written as

ln 119871 (Θ | 119884) = minus12 (ln (2120587) + 1) 119870sum119896=1

119873119896sum119894=1

119872119894119896minus 12 ln (2120578) 119870sum

119896=1

119873119896sum119894=1

119872119896119894minus 12 119870sum119896=1

119873119896sum119894=1

10038161003816100381610038161003816Σ11989611989410038161003816100381610038161003816 (25)

The MLE of 120573 and 2120576 can be obtained by maximizingthe profile log-likelihood function in (25) through a three-dimensional search Then substitute them into (24) we canobtain MLE for 120583120578 and 1205902120578

The value of and 2120576 can be obtained by the followingequations

= 120581 sdot 21205782120576 = 1205902120576 sdot 2120578 (26)

42 Parameter Estimation of SSADT The degradation pro-cess of SSADT shown as (18) is relatively complicated Thuswe introduce the accelerated factor first for the sake ofsimplicity The variable 120577119896 is represented by the product of 1205771and the accelerated factor 1205721198961

120589119896 = 12058911205721198961 (27)

The SSADP can be rewritten as

119909119896119894119895

=

12057811989412058911205961119894119895 + radic1205811205891119861 (12059612119895) 1205911198940 le 120596119894119896119895 le 12059111989411205781198941205891 (1205911119894 + 12057221 (1205962119894119895 minus 1205911119894)) + radic1205811205891119861 (1205911119894 + 12057221 (1205962119894119895 minus 1205911119894)) 1205911119894 le 1205962119894119895 le 1205912119894sdot sdot sdot1205781198941205891(119896minus1sum

119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119896119894119895 minus 120591(119896minus1)119894)) + radic1205811205891119861(119896minus1sum119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119896119894119895 minus 120591(119896minus1)119894)) 120591(119896minus1)119894 le 120596119896119894119895 le 120591119896119894sdot sdot sdot1205781198941205891(119870minus1sum

119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119870119894119895 minus 120591(119870minus1)119894)) + radic1205811205891119861(119870minus1sum119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119870119894119895 minus 120591(119870minus1)119894)) 120591(119870minus1)119894 le 120596119870119894119895 le 120591119870119894

(28)

Specify 120594119894119896119895 = sum119896minus1119899=1 1205721198991(120591119894119899 minus 120591119894(119899minus1)) + 12057221(120596119894119896119895 minus 120591119894(119896minus1))then the SSADP can be expressed a general formula asfollows

119909119896119894119895 = 1205781198941205891120594119896119894119895 + radic1205811205891119861 (120594119896119894119895) 120591(119896minus1)119894 le 120596119896119894119895 ge 120596119896119894119896 = 1 119870 119894 = 1 119873 119895 = 1 119872119896119894(29)

The foregoing transformation is equivalent to convertingthe degradation driven by stress 119878119896 to the degradation understress 1198781 in physics and the converted equivalent time is 120594119896119894119895At the moment the matrix of degradation and equivalenttransformed time are still bidimensional Because of thecontinuity of the degradation process the matrix can bewritten as a column vector with119872119894 element where119872119894 equalsthe sum of119872119894119896 119896 = 1 119870 and the subscript can be written

Mathematical Problems in Engineering 7

as 119898 for the 119894th product 119898 = 1 119872119894 Then the observeddegradation can be expressed as

119910119894119898 = 119909119894119898 + 120576119894119898119909119894119898 = 1205781198941205891120594119894119898 + 1205811205891119861 (120594119894119898) 1 le 119898 le 119870sum

119896=1

119872119894119896(30)

Define Δ1205941198941 = 1205941198941 Δ120594119894119898 = 120594119894119898 minus 120594119894(119898minus1) Δ120594119894 =(Δ1205941198941 Δ120594119894119872119894)1015840 119877119894 = Δ120594119894 and Δ119910119894119898 = 119910119894119898 minus 119910119894(119898minus1) for119894 = 1 119873 119898 = 1 119872119894 Then Δ119910119894 = (Δ1199101198941 Δ119910119894119872119894)1015840follows a multivariate normal distribution 119873(1205831205781205891119877119894 Π119894)where

Π119894 = 119882119894 + 1205902120578120589211198771198941198771015840119894 (119882119901119902)119894 = cov (Δ119910119894119901 Δ119910119894119902)

=

1205811205891119877119894119901 + 1205902120576 119901 = 119902 = 11205811205891119877119894119901 + 21205902120576 119901 = 119902 gt 1minus1205902120576 119901 = 119902 + 1 or 119901 = 119902 minus 10 otherwise

(31)

So the log-likelihood function can be expressed as

ln 119871 (Θ | 119884)= minus12 ln (2120587) 119873sum

119894=1

119872119894 minus 12 119873sum119894=1

ln (1003816100381610038161003816Π1198941003816100381610038161003816)minus 12 119873sum119894=1

(Δ119910119894 minus 1205831205781205891119877119894)1015840Πminus1119894 (Δ119910119894 minus 1205831205781205891119877119894) (32)

where1003816100381610038161003816Π1198941003816100381610038161003816 = 10038161003816100381610038161198821198941003816100381610038161003816 (1 + 1205902120578120589211198771015840119894119882minus1119894 119877119894)Πminus1119894 = 119882minus1119894 minus 1205902120578120589211 + 1205902120578120589211198771015840119894119882minus1119894 119877119894119882minus1119894 1198771198941198771015840119894119882minus1119894 (33)

Taking the first partial derivatives of the log-likelihoodfunction with respect to 120583120578 1205902120578 yields

120597 ln 119871 (Θ | 119884)120597120583120578 = 1205891 119873sum119894=1

1198771015840119894Πminus1119894 Δ119910119894 minus 12058312057812058921 119873sum119894=1

1198771015840119894Πminus1119894 119877119894 (34)

120597 ln 119871 (Θ | 119884)1205971205902120578 = minus 119873sum119894=1

120589211198771015840119894119882minus1119894 1198771 + 1205902120578120589211198771015840119894119882minus1119894 119877119894+ 119873sum119894=1

12058921 (Δ119910119894 minus 1205831205781205891119877119894)1015840119882minus1119894 1198771198941198771015840119894119882minus1119894 (Δ119910119894 minus 1205831205781205891119877119894)(1 + 1205902120578120589211198771015840119894119882minus1119894 119877)2(35)

For the special value of (120581 120573 1205902120576 ) setting the derivation of120597 ln 119871(Θ | 119884)120597120583120578 to zero the restricted MLE for 120583120578 can beexpressed as

120578 = sum119873119894=1 1198771015840119894Πminus1119894 Δ1199101198941205891sum119873119894=1 1198771015840119894Πminus1119894 119877119894 (36)

Similarly set the derivation of 120597 ln 119871(Θ | 119884)120597120583120578 to zeroand the results of the MLE for 1205902120578 will be discussed on twocases

Case 1 It was assumed that the number of the measurementsand the measurement points of each sample are the same forall of the samples under all of the accelerated stressThat is tosay the subscript of 119877119894 Π119894 and119882119894 can be removed

Thus the restricted MLE for 120583120578 can be expressed as

120578 = sum119873119894=1 1198771015840Πminus1119894 Δ11991011989412058911198731198771015840Πminus1119894 119877 (37)

The first partial derivatives of the log-likelihood functionto 1205902120578 can be rewritten as

120597 ln 119871 (Θ | 119884)1205971205902120578 = minus119873 120589211198771015840119882minus11198771 + 1205902120578120589211198771015840119882minus1119877+ 12058921(1 + 1205902120578120589211198771015840119882minus1119877)2sdot 119873sum119894=1

(Δ119910119894 minus 1205831205781205891119877)119882minus11198771198771015840119882minus1 (Δ119910119894 minus 1205831205781205891119877)1015840 (38)

For the special value of (120581 120573 1205902120576 ) setting the derivation of120597 ln 119871(Θ | 119884)1205971205902120578 to zero the result of the MLE for 1205902120578 can beexpressed as2120578

= sum119873119894=1 (Δ119910119894 minus 1205831205781205891119877)119882minus11198771198771015840119882minus1 (Δ119910119894 minus 1205831205781205891119877)101584011987312058921 (1198771015840119882minus1119877)2minus 1120589211198771015840119882minus1119877

(39)

Substituting (37) (39) into (32) and simplifying the profilelog-likelihood function can be written as

ln 119871 (120581 120573 1205902120576 | 120578 2120578 119884) = minus1198721198732 ln (2120587) minus 1198732 minus 1198732sdot ln (|119882|)minus 12

119873sum119894=1

Δ1199101015840119894119882minus1Δ119910119894 minus sum119873119894=1 (1198771015840119882minus1Δ119910119894)21198771015840119882minus1119877 minus 12sdot ln

sum119873119894=1 (1198771015840119882minus1Δ119910119894)21198731198771015840119882minus1119877 minus sum119873119894=1 (1198771015840119882minus1Δ119910119894)211987321198771015840119882minus1119877

(40)

8 Mathematical Problems in Engineering

where119872 is themeasurement time point of each samplewhichis the same for all of the samples based on the assumption ofCase 1

The MLE of 120581 120573 and 1205902120576 can be obtained by maximizingthe profile log-likelihood function in (40) through a three-dimensional searchThen substitute them into (37) and (39)we can obtain the MLE for 120583120578 and 1205902120578 Case 2 The number of the measurements and the measure-ment points of each sample are different for all of the samplesunder all of the accelerated stress In this case the first partialderivatives of the log-likelihood function to 1205902120578 are shown asin (35)There may be no analytical form by setting the partialderivatives to zero The evaluation of 120583120578 can be expressed as(36) Then the profile log-likelihood function of 1205902120578 120581 120573 1205902120576can be written as

ln 119871 (120581 120573 120590120576 1205902120578 | 120578 119884)= minus12 ln (2120587) 119873sum

119894=1

119872119894 minus 12 119873sum119894=1

ln (1003816100381610038161003816Π1198941003816100381610038161003816)minus 119873sum119894=1

(Δ119910119894 minus 1205781205891119877119894)Πminus1119894 (Δ119910119894 minus 1205781205891119877119894)1015840 (41)

The MLE of 1205902120578 120581 120573 and 1205902120576 can be obtained by maxi-mizing the profile log-likelihood function in (41) through afour-dimensional search Then substitute them into (36) wecan obtain the MLE for 120583120578

It is not to say that we can only use degradation forCSADT and increment for SSADT but just make an intro-duction to both of the two methods in the limited space Inaddition we could verify the results by comparing the esti-mation calculated by the two methods to avoid computationerrors

5 Simulation Data Analysis

In order to validate the model described before and theparameter estimation methods simulation test was con-ducted

51 The Simulation Method of CSADT Data The parametersΘ = 120583120578 1205902120578 120581 120573 1205902120576 should be given before the simulationand simulation process is shown as in the following steps

Step 1 Set 119896 = 1Step 2 Generate 119873119894 random numbers with subscripts 119894 119894 =1 2 119873119896 subject to normal distribution with mean 120583120578 andvariance 1205902120578 Step 3 Let 119897 = 0Step 4 Calculate119883119896119894(119897+1) using the Euler approximation [36]

119883119896119894(119897+1) = 119883119896119894119897 + 120578119894120589119896120596 (Δ119905) + 120581120589119896radic120596 (Δ119905)Ψ (42)

where Ψ is the random number according to the standardnormal distribution and Δ119905 is the step size in simulation and1198831198961198940 = 0 As such set 119897 = 119897 + 1 and return to Step 4 to get thedegradation path until 119897 = 119871119896119894+1 where 119871119896119894 is the simulationsteps for 119894th under stress 119878119896Step 5 Set 119896 = 119896 + 1 and return to Step 2 until 119896 = 119870 + 1Step 6 Extract the degradation based on the predefinedmeasurement time point 119905119896119894119895 where 119896 = 1 2 119870 119894 =1 2 119873119896 and 119895 = 1 2 119872119896119894 from the data set 119883119896119894119897 andget the data set of degradation119883119896119894119895Step 7 Simulate the measured degradation 119884119896119894119895 = 119883119896119894119895 + 120576119896119894119895where 120576119896119894119895 sim 119873(0 1205902120576 )52 The Simulation Method of SSADT Data There are somedifferences in the simulation process for SSADT comparedwith CSADT The simulation process is shown as follows

Step 1 Generate 119873 random numbers with subscripts 119894 119894 =1 2 119873 subject to normal distribution with mean 120583120578 andvariance 1205902120578 Step 2 Let 119896 = 1Step 3 Set 119897 = 1Step 4 Calculate119883119896119894(119897+1)

119883119896119894(119897+1) = 119883119896119894119897 + 120578119894120589119896120596 (Δ119905) + 120581120589119896radic120596 (Δ119905)Ψ (43)

where Ψ is the random number according to the standardnormal distribution and Δ119905 is the step size in simulation and1198831198961198940 = 01198831198961198940 = 119883(119896minus1)119894119871(119896minus1) (119896 gt 1) As such set 119897 = 119897+1 andreturn to Step 4 to get the degradation path until 119897 = 119871119896119894 + 1Step 5 Set 119896 = 119896 + 1 and return to Step 2 until 119896 = 119870 + 1Step 6 Extract the degradation based on the predefinedmeasurement time point 119905119896119894119895 where 119896 = 1 2 119870 119894 =1 2 119873 and 119895 = 1 2 119872119894119896 from the data set 119883119896119894119897 andget the data set of degradation119883119896119894119895Step 7 Simulate the measured degradation 119884119896119894119895 = 119883119896119894119895 + 120576119896119894119895where 120576119896119894119895 sim 119873(0 1205902120576 )53 The Analysis of the Simulated SSADT Data We justgive the analysis of simulated SSADT data here becausewe would give a case study of the CSADT later It wasassumed that the accelerated stress is temperature and thesimulation test contains 4 stresseswhich are 50∘C 60∘C 70∘Cand 80∘C For simplicity the transformed time function isset as Λ(119905) = 119905 and 20 degradation paths are generatedThen the parameters for degradation process are assumedas Θ = 5 4 025 minus3000 001 Moreover the time intervalmeasurement is 25 h for each sample of the whole simulationtest with 40 measurement time points The degradation dataare depicted as in Figure 1

Mathematical Problems in Engineering 9

Table 2 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 log-LF AIC RETruth value 5 1 025 mdash minus3000 001 mdash mdash mdash1198720 501 139 028 mdash minus297561 00078 142228 minus283456 084021198721 512 175 mdash 12907 lowast 10minus4 minus298081 00129 112289 minus223577 562711198722 620 mdash mdash 15213 lowast 10minus4 minus305175 13789 lowast 10minus9 109361 minus217721 78049

0 200 400 600 800 1000

0

05

1

15

2

25

th

y

Figure 1 The simulation degradation paths of SSADT

For simplicity the degradation model for SSADT pro-posed in this paper is referred to as1198720 the model presentedby Tang et al [17] as 1198721 and the stochastic SSADT modeldescribed in reference [13] as 1198722 The estimation results ofthe unknown parameters the log-likelihood function valueand the Akaike information criterion (AIC) [37] which isevaluated by (44) are summarized in Table 2 As we cansee from Table 2 our model clearly outperforms model interms of the log-LF and AIC The estimates of 120583120578 1205902120578 and1205902120576 are more close to the truth values compared with 1198721and 1198722 Meanwhile the diffusion-related variables includeparameters 120581 and 120573 and the diffusion parameter under thefour stresses can be obtained by (10) which is (28377 lowast 10minus548534 lowast 10minus5 78244 lowast 10minus5 12001 lowast 10minus4) in 1198721 and(22334lowast10minus5 39912lowast10minus5 64597lowast10minus5 99423lowast10minus5) forthe true model Moreover taking 1198781 as a reference a criterionformulizing the relative error (RE) is shown as (45) It is clearthat1198720 has the smallest RE from Table 2

AIC (119873) = minus2 ln 119871 (Θ) + 2119873 (44)

RE = 1003816100381610038161003816100381610038161003816100381610038161003816120583120578 minus 120578120578

1003816100381610038161003816100381610038161003816100381610038161003816 +100381610038161003816100381610038161003816100381610038161003816100381610038161205902120578 minus 21205782120578

10038161003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161003816(1205902119861)1 minus (2119861)1(2119861)1

10038161003816100381610038161003816100381610038161003816100381610038161003816+ 1003816100381610038161003816100381610038161003816100381610038161003816120573 minus

1003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161205902120576 minus 21205762120576

1003816100381610038161003816100381610038161003816100381610038161003816 (45)

It is assumed that the failure threshold119908 is equal to 5 andthe normal stress is 25∘C The PDF and CDF under normalstress are shown as in Figure 2 The phenomenon can beobserved where the PDF and CDF of our model most nearlyapproached the real model By contrasting 1198721 with 1198722 itcan be concluded that the neglect of random effects wouldresult in sharper PDF which is consistent with the result inreference [17] And it is noteworthy that the 05 quantiles ofthe failure life are almost equal from the CDF of the threemodels But there is a greater difference between the realmodel and 1198721 while the indicator is taken as mean time tofailure as well as the difference between the real model and1198722 So the correlation between the stress and the diffusionparameter should not be neglected Next we are going toprove the superiority of our model in practical application

6 Case Application

The CSADT model with covariates and random effects isverified by the accelerated degradation data of carbon-filmresistors whose raw data set is explicitly given in Table C3of Meeker and Escobar [38]There were 29 samples and threeaccelerated temperature stresses in the whole test where ninesamples were observed at 83∘C the remaining two stresses133∘C and 173∘C had 10 samples for each stress All of thesamples were observed at the same time points with 1199050 = 01199051 = 452 1199052 = 1030 1199053 = 4341 and 1199054 = 8084 (inhours) Similar to [39] it was assumed that the standardoperating temperature was 50∘C and the threshold value forpercent increase in resistance was taken to be 119862 = 12 Itis reasonable and computationally easier to use the ratios inresistance to the initial value for each rather than the percentincrease [9] For ease of calculation the logarithm of the ratiowas taken as the degradation here That is the value 028of the percent increase was changed to 2796119864 minus 3 and soon and the transformed threshold value 119908 was 01133 Thedegradation paths of all the sample were as shown in Figure 3It can be seen that the degradation of the samples uniformlyshowed a nonlinear characteristic especially at the beginningof the ADT Thus the degradation is modeled with the timescale transformedWiener process as (1) with 120596 = Λ(119905) = 119905119887

Similarly the proposed degradation model for SSADT inthis paper is referred to as1198720 themodel presented by Tang etal [17] as1198721 and themodel described in reference [9] as1198723The estimation results of the unknown parameters the log-likelihood function value and the AIC are shown in Table 3The estimated parameters 119887 are approximately equal for thethree models Take 119887 = 050 as an example to show the time

10 Mathematical Problems in Engineering

Table 3 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 b Log-LF AIC MTTF1198720 823 207 108119864 minus 02 mdash minus420282 182119864 minus 20 050 51848 minus102497 380119864 + 071198721 1471 433 mdash 413119864 minus 07 minus458656 506119864 minus 20 053 48637 minus96073 515119864 + 071198722 1111 mdash mdash 676119864 minus 07 minus447982 411119864 minus 18 053 46837 minus92674 406119864 + 07

0 1 2 3 4 50

05

1

15

2

25

th

The P

DF

Real model

times104

times10minus4

M1

M0

M2

0 1 2 3 4 50

02

04

06

08

1

th

The C

DF

times104

Real modelM1

M0

M2

Figure 2 The comparison of the PDF and CDF of the three models for simulated SSADT data

0 2000 4000 6000 80000

0002

0004

0006

0008

001

th

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 2000 4000 6000 80000

001

002

003

th

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 2000 4000 6000 8000th

0

002

004

006

008

01

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 3 The degradation paths of carbon-film resistors

scale transformed degradation paths as in Figure 4 which areapproximately straight lines compared with Figure 3

Our model has the largest log-LF and smallest AICcompared with1198721 and1198722 The result proves that our modelis more suitable for the degradation data Besides 1198721 hasthe larger log-LF and smaller AIC compared with1198722 So therandom effects could not be neglected It can be observed thatthe variance of the drift parameter is slightly largerwhichmaybe owing to the fixed value of diffusion parameter in1198721

The PDF and CDF under the standard operating temper-ature are as shown in Figure 5The time corresponding to thepeak values of PDF were as follows 2676 lowast 107 1049 lowast 107

and 5028lowast106 for1198720 to1198722 But theMTTF of1198720 isminimalfrom Table 3 And the uncertainty in the estimated PDFs ofthe lifetimes under1198720 is smallest compared to1198721 and1198722 asseen in Figure 5When time 119905 approaches positive infinity theCDF of ourmodel which is calculated by (13) tends illimitablyto 1 and the CDF of 1198721 is roughly equal to 09997 It followsthat our model could do well in distinguishing the differencefrom random effects and the effect of covariates Anotherthing which is worth noting is that the measurement error isquite tiny in the case but it is just an individual phenomenonThemeasurement error should also be considered in practicalapplication

Mathematical Problems in Engineering 11

0 20 40 60 800

0002

0004

0006

0008

001

Λ(t)

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 20 40 60 800

001

002

003

Λ(t)

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 20 40 60 800

002

004

006

008

01

Λ(t)

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 4 The degradation paths of carbon-film resistors under transformed time scale

0 2 4 6 8 100

05

1

15

2

25

3

35

th

times10minus8

M0

M1

M2

The P

DF

times107

0 2 4 6 8 100

02

04

06

08

1

th

M0

M1

M2

The C

DF

times107

Figure 5 The comparison of the PDF and CDF of the three models for real CSADT data

7 Conclusion

In this paper the degradation models based on nonlinearWiener process are established for both constant stress accel-erated degradation data and step stress accelerated degrada-tion data Before the establishment the relationship betweenthe drift parameter and stress variables is derived based onthe invariance principle of failure mechanism and Nelsonassumption so is the relationship between the diffusionparameter and stress variables It is concluded that the ratioof drift parameters under two stresses is a constant which isirrelevant to the testing time and depends only on the twostresses as long as the ratio of diffusion parameters is equalto the ratio of drift parameters And the ratio is defined asaccelerated factor Besides the random effects are also takeninto consideration where the drift parameter is assumed to be

normally distributed and the diffusion parameter is same forall of the samples under a certain stress Then the PDF andCDF of the FHT are deduced considering random effects

Because of the dependency between the diffusion param-eter and stress variables the degradation process is quitedifferent either for CSADT or for SSADT The CSADP andSSADP with random effects are modeled Moreover theunknown parameters are solved by MLE based on the twoproperties of Wiener process At the end of the paper thesimulated data of SSADT and the CSADTdata of carbon-filmresistors are both analyzed to verify the proposed model Itis concluded that the model has the biggest log-LF and thesmallest AIC compared with the two other models

The innovation of this paper lies in the following Firstthe random effects are considered under the new relationshipbetween the diffusion parameter and accelerated stresses

12 Mathematical Problems in Engineering

Second the degradation process was modeled for bothCSADT and SSADT Thirdly the unknown parameters wereestimated based on the two properties of Wiener process andthe result of theMLE for 1205902120578 is discussed on two cases Fourththe measurement error of the degradation data is also valued

However we have only considered the random effects ofthe drift parameter in this paper due to the complexity of thecomputation A further research may consider the randomeffects of the diffusion parameter into the model At the sametime the study of the paper may provide new ideas for therelativity analysis between the parameters of other stochasticprocess and stress variables

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by Jiangsu Province GraduateStudent Scientific Research Innovation Project of China(Project KYLX15 0330)The help is gratefully acknowledged

References

[1] H-F Yu ldquoOptimal classification of highly-reliable productswhose degradation paths satisfyWiener processesrdquo EngineeringOptimization vol 35 no 3 pp 313ndash324 2003

[2] XWang N Balakrishnan and B Guo ldquoResidual life estimationbased on a generalized Wiener degradation processrdquo ReliabilityEngineering and System Safety vol 124 pp 13ndash23 2014

[3] Y Ren Q Feng T Ye and B Sun ldquoA novel model of reliabilityassessment for circular electrical connectorsrdquo IEEE Transac-tions on Components Packaging andManufacturing Technologyvol 5 no 6 pp 755ndash761 2015

[4] Z-S Ye and M Xie ldquoStochastic modelling and analysis ofdegradation for highly reliable productsrdquo Applied StochasticModels in Business and Industry vol 31 no 1 pp 16ndash32 2015

[5] H Lim and B-J Yum ldquoOptimal design of accelerated degrada-tion tests based on Wiener process modelsrdquo Journal of AppliedStatistics vol 38 no 2 pp 309ndash325 2011

[6] C J Lu and W Q Meeker ldquoUsing degradation measures toestimate a time-to-failure distributionrdquo Technometrics vol 35no 2 pp 161ndash174 1993

[7] G AWhitmore ldquoEstimating degradation by a wiener diffusionprocess subject to measurement errorrdquo Lifetime Data Analysisvol 1 no 3 pp 307ndash319 1995

[8] Z Pan and N Balakrishnan ldquoReliability modeling of degra-dation of products with multiple performance characteristicsbased on gamma processesrdquo Reliability Engineering amp SystemSafety vol 96 no 8 pp 949ndash957 2011

[9] C Park and W J Padgett ldquoAccelerated degradation modelsfor failure based on geometric Brownian motion and gammaprocessesrdquo Lifetime Data Analysis vol 11 no 4 pp 511ndash5272005

[10] Z-S Ye and N Chen ldquoThe inverse Gaussian process as adegradation modelrdquo Technometrics vol 56 no 3 pp 302ndash3112014

[11] A Onar and W J Padgett ldquoAccelerated test models with theinverse Gaussian distributionrdquo Journal of Statistical Planningand Inference vol 89 no 1-2 pp 119ndash133 2000

[12] T-R Tsai Y L Lio and N Jiang ldquoOptimal decisions on theaccelerated degradation test plan under the Wiener processrdquoQuality Technology and Quantitative Management vol 11 no4 pp 461ndash470 2014

[13] C-M Liao and S-T Tseng ldquoOptimal design for step-stressaccelerated degradation testsrdquo IEEE Transactions on Reliabilityvol 55 no 1 pp 59ndash66 2006

[14] L C Tang G Yang and M Xie ldquoPlanning of step-stress accel-erated degradation testrdquo in Proceedings of the Annual Reliabil-ity and Maintainability SymposiummdashProceedings InternationalSymposium on Product Quality and Integrity pp 287ndash292January 2004

[15] C-H Hu M-Y Lee and J Tang ldquoOptimum step-stress accel-erated degradation test for Wiener degradation process underconstraintsrdquo European Journal of Operational Research vol 241no 2 pp 412ndash421 2015

[16] S Tang ldquoStep stress accelerated degradation process modelingand remaining useful life estimationrdquo Journal of MechanicalEngineering vol 50 no 16 p 33 2014

[17] S Tang X Guo C Yu H Xue and Z Zhou ldquoAccelerated degra-dation tests modeling based on the nonlinear wiener processwith random effectsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 560726 11 pages 2014

[18] L Wang R Pan X Li and T Jiang ldquoA Bayesian reliability eval-uation method with integrated accelerated degradation testingand field informationrdquo Reliability Engineering amp System Safetyvol 112 pp 38ndash47 2013

[19] Z-S Ye N Chen and Y Shen ldquoA new class of Wiener processmodels for degradation analysisrdquo Reliability Engineering ampSystem Safety vol 139 pp 58ndash67 2015

[20] G A Whitmore and F Schenkelberg ldquoModelling accelerateddegradation data using wiener diffusion with a time scaletransformationrdquo Lifetime Data Analysis vol 3 no 1 pp 27ndash451997

[21] K A Doksum and A Hoyland ldquoModels for variable-stressaccelerated life testing experiments based on Wiener processesand the inverse Gaussian distributionrdquo Technometrics vol 34no 1 pp 74ndash82 1992

[22] H Liao and E A Elsayed ldquoReliability inference for field con-ditions from accelerated degradation testingrdquo Naval ResearchLogistics vol 53 no 6 pp 576ndash587 2006

[23] H-WWang T-X Xu andW-YWang ldquoRemaining life predic-tion based on wiener processes with ADT prior informationrdquoQuality and Reliability Engineering International vol 32 no 3pp 753ndash765 2015

[24] C-Y Peng ldquoInverse Gaussian processes with random effectsand explanatory variables for degradation datardquo Technometricsvol 57 no 1 pp 100ndash111 2015

[25] X Pan XHuang Y Chen YWang and R Kang ldquoConnotationof failure mechanism consistency and identificationmethod foraccelerated testingrdquo in Proceedings of the Prognostics and SystemHealth Management Conference (PHM-Shenzhen rsquo11) May 2011

[26] W B Nelson Accelerated Testing Statistical Models Test Plansand Data Analysis John Wiley amp Sons 2009

[27] Z-S Ye Y Hong and Y Xie ldquoHow do heterogeneities in oper-ating environments affect field failure predictions and testplanningrdquo The Annals of Applied Statistics vol 7 no 4 pp2249ndash2271 2013

Mathematical Problems in Engineering 13

[28] Z-X Zhang X-S Si C-H Hu Q Zhang T Li and C XuldquoPlanning repeated degradation testing for products with three-source variabilityrdquo IEEE Transactions on Reliability vol 65 no2 pp 640ndash647 2016

[29] C-Y Peng and S-T Tseng ldquoMis-specification analysis of lineardegradation modelsrdquo IEEE Transactions on Reliability vol 58no 3 pp 444ndash455 2009

[30] X-S Si W Wang C-H Hu D-H Zhou and M G PechtldquoRemaining useful life estimation based on a nonlinear diffu-sion degradation processrdquo IEEE Transactions on Reliability vol61 no 1 pp 50ndash67 2012

[31] X-S Si W Wang C-H Hu M-Y Chen and D-H Zhou ldquoAWiener-process-based degradationmodel with a recursive filteralgorithm for remaining useful life estimationrdquo MechanicalSystems and Signal Processing vol 35 no 1-2 pp 219ndash237 2013

[32] C-C Tsai S-T Tseng and N Balakrishnan ldquoMis-specificationanalyses of gamma andWiener degradation processesrdquo Journalof Statistical Planning and Inference vol 141 no 12 pp 3725ndash3735 2011

[33] F Haghighi and S J Bae ldquoReliability estimation from lineardegradation and failure time data with competing risks under astep-stress accelerated degradation testrdquo IEEE Transactions onReliability vol 64 no 3 pp 960ndash971 2015

[34] Z-S Ye Y Wang K-L Tsui and M Pecht ldquoDegradation dataanalysis usingwiener processeswithmeasurement errorsrdquo IEEETransactions on Reliability vol 62 no 4 pp 772ndash780 2013

[35] J Huang D S Golubovic S Koh et al ldquoDegradation modelingof mid-power white-light LEDs by using Wiener processrdquoOptics Express vol 23 no 15 pp A966ndashA978 2015

[36] A Beskos O Papaspiliopoulos G O Roberts and P Fearn-head ldquoExact and computationally efficient likelihood-basedestimation for discretely observed diffusion processes (withdiscussion)rdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 68 no 3 pp 333ndash382 2006

[37] J Shang and J E Cavanaugh ldquoAn assumption for the develop-ment of bootstrap variants of the Akaike information criterionin mixed modelsrdquo Statistics and Probability Letters vol 78 no12 pp 1422ndash1429 2008

[38] W QMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[39] W J Padgett andM A Tomlinson ldquoInference from accelerateddegradation and failure data based on Gaussian process mod-elsrdquo Lifetime Data Analysis vol 10 no 2 pp 191ndash206 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

drift parameter 120582 and the diffusion parameter 120590119861 are bothconsidered as the randomparameters So like Peng and Tseng[29] Si et al [30 31] and Tsai et al [32] it is also assumedthat different units have different drift parameters while alldiffusion parameters have the same value under a certain

stress Then we have 120578 sim 119873(120583120578 1205902120578) where 120583120578 and 1205902120578 are themean and variance of the parameter 120578 separately

Considering the random effects and the effects of accel-erated stresses on the drift parameter 120582 and the diffusionparameter 120590119861 we have the PDF and CDF of FHT under stress119878119896 as

119891119878119896 (119905) = 119908radic21205871205963 (120581120589119896 + 12059021205781205892119896120596) exp(minus (119908 minus 120583120578120589119896120596)22120596 (120581120589119896 + 12059021205781205892119896120596)) 119889120596119889119905 (12)

119865119878119896 (119905) =Φ( 120583120578120589119896120596 minus 119908radic120596(120581120589119896 + 12059021205781205892119896120596)) + exp(2120583120578119908120581 + 2120590212057811990821205812 ) sdot Φ(minus21205902120578120589119896119908120596 + 120581 (120583120578120589119896120596 + 119908)120581radic120581120589119896120596 + 120590212057812058921198961205962 ) if 119905 lt infinΦ(120583120578120590120578) + exp(2120583120578119908120581 + 2120590212057811990821205812 )Φ(minus21205902120578119908 + 120581120583120578120581120590120578 ) if 119905 997888rarr infin

(13)

whereΦ(sdot) is the distribution function of the standard normaldistribution

When 120583120578120590120578 approach infinity the transformed FHTunder stress 119878119896 is subject to inverse Gaussian (IG) distribu-tion

3 Model the CSADP and SSADP withRandom Effects

As themost-usedADTCSADTand SSADThave beenwidelyresearched But the models are quite different while the driftparameter 120582 and the diffusion parameter 120590119861 all depend onstress variables especially for SSADT

31 Modeling the CSADP with Random Effects Let 1198781 119878119896 119878119870 denote119870 stress level higher than normal stresslevel 1198780 such that 1198781 lt sdot sdot sdot lt 119878119870 and the subscript 119896 iscorresponding to the 119896th stress level 119896 = 1 2 119870 Supposethat there are 119873119896 units of samples tested under a constantaccelerated stress 119878119896 and each sample is measured 119872119896119894 timesat the 119896th stress level 119894 = 1 2 119873119896 The degradationat transformed time 120596119896119894119895 where 120596119896119894119895 = Λ(119905119896119894119895) is 119909119896119894119895 119895 =1 2 119872119896119894 The degradation process can be formulated as

119909119896119894119895 (120596119896119894119895 | 119878119896) = 120578119894120589119896120596119896119894119895 + radic120581120589119896119861 (120596119896119894119895) (14)

32Modeling the SSADPwith RandomEffects Similarly withCSADT it was assumed that there are 119870 accelerated stresslevels 1198781 lt sdot sdot sdot 119878119896 lt sdot sdot sdot lt 119878119870 in the whole test But thenumber of the samples is only 119873 which is different fromsum119870119896=1119873119896 inCSADT that verifies the characteristic that SSADTneeds fewer samples compared with CSADT [33] Supposethat each sample is also measured 119872119896119894 times at the 119896thstress level 119894 = 1 2 119873 119896 = 1 2 119870 And 120591119896119894 is thetransformed time scale where 120591119896119894 = Λ(119905119896119894119872119896119894) at which thestress changes from the 119896th stress level to the (119896 + 1)th stresslevel of the 119894th sample 119896 = 1 2 119870 minus 1 Besides 1205911198940 equals

0 and 120591119894119870 is the transformed end time of the 119870-step stressaccelerated test of the 119894th sample

The degradation process for CSADT is the same asSSADT under accelerated stress 1198781

1199091119894119895 = 12057811989412058911205961119894119895 + radic1205811205891119861 (1205961119894119895) 0 le 1205961119894119895 le 1205911119894 (15)

Raise the accelerated stress up to 1198782 at transformed timepoint 1205911119894 for the 119894th product Then the degradation is drivenby the accelerated stress 1198782 and the corresponding parameterscan be obtained by (9) are (10) where the drift parameter andthe diffusion parameter are equal to 1205781198941205892 and 1205811205892 separatelyThus the degradation consisted of the degradation driven by1198781 and the degradation driven by 1198782

1199092119894119895 = 12057811989412058911205911198941 + radic1205811205891119861 (1205911198941) + 1205781198941205892 (1205961198942119895 minus 1205911198941)+ radic1205811205892119861 (1205961198942119895 minus 1205911198941) (16)

Similarly the accelerated stress is turning up to 1198783 attransformed timepoint 1205912119894 and the degradation can bewrittenas

1199093119894119895 = 2sum119899=1

[120578119894120589119899 (120591119894119899 minus 120591119894(119899minus1)) + radic120581120589119899119861 (120591119894119899 minus 120591119894(119899minus1))]+ 1205781198941205893 (1205961198943119895 minus 1205911198942) + radic1205811205893119861 (1205961198943119895 minus 1205911198942)

(17)

Mathematical Problems in Engineering 5

According to the analysis the degradation process ofSSADT can be formulated as

119909119896119894119895 =

1205781198941205891120596119894119896119895 + radic1205811205891119861 (120596119894119896119895) 1205911198940 le 120596119894119896119895 le 120591119894112057811989412058911205911198941 + radic1205811205891119861 (1205911198941) + 1205781198941205892 (120596119894119896119895 minus 1205911198941) + radic1205811205892119861 (120596119894119896119895 minus 1205911198941) 1205911198941 le 120596119894119896119895 le 1205911198942sdot sdot sdot119896minus1sum119899=1

[120578119894120589119899 (120591119894119899 minus 120591119894(119899minus1)) + radic120581120589119899119861 (120591119894119899 minus 120591119894(119899minus1))] + 120578119894120589119896 (120596119894119896119895 minus 1205911198942) + radic120581120589119896119861 (120596119894119896119895 minus 1205911198942) 120591119894119896minus1 le 120596119894119896119895 le 120591119894119896sdot sdot sdot119870minus1sum119899=1

[120578119894120589119899 (120591119894119899 minus 120591119894(119899minus1)) + 120581120589119899119861 (120591119894119899 minus 120591119894(119899minus1))] + 120578119894120589119870 (120596119894119896119895 minus 1205911198942) + 120581120589119870119861 (120596119894119896119895 minus 1205911198942) 120591119894119870minus1 le 120596119894119896119895 le 120591119894119870

(18)

4 Parameter Estimation

In real applications it is inevitable that some measurementerrors may be introduced during the observation process[34] When a measurement is taken the observed degrada-tion is shown as follows for both CSADT and SSADT

119910119896119894119895 (120596119896119894119895 | 119878119896) = 119909119896119894119895 (120596119896119894119895 | 119878119896) + 120576119896119894119895119896 = 1 119870 119894 = 1 119873119896 119895 = 1 119872119896119894 (19)

where the measurement errors 120576119896119894119895 are assumed to be iidrealizations of 120576119896119894119895 sim 119873(0 1205902120576 ) and mutually independent of119909119896119894119895 [7 34] and 119873119896 is equal to 119873 for all of the acceleratedstresses of SSADT

The unknown parameters in the models are Θ =120583120578 1205902120578 120581 120573 1205902120576 There are two methods to deal with theunknown parameters with each considering one of thetwo main properties of Brownian motion which could becharacterized as follows [35]

(P1) The increment Δ119909(120596) = 119909(120596 + Δ120596) minus 119909(120596) isindependent of the degradation 119909(120596) which meansthat if 0 le 1205961 le 1205962 le 1205963 le 1205964 then 119909(1205962) minus119909(1205961) and 119909(1205964) minus 119909(1205963) are independent randomvariables Meanwhile the increment Δ119909(120596) follows anormal distribution where the mean equals 120582Δ120596 andthe variance is 1205902119861Δ120596

(P2) According to (P1) the degradation119909(120596) follows a nor-mal distribution with the mean 120582120596 and the variance1205902119861120596

Even though our models concern the unit-to-unitevariability the essence of the Wiener process remains thesame Owing to space constraints this paper deals with theunknown parameter based on property (P2) in the case ofCSADT and property (P1) in the case of SSADT

41 Parameter Estimation of CSADT Specify 119879119896119894 = (1198791198961198941 119879119896119894119872119896119894)1015840 119879119896119894119895 = 120596119896119894119895 119884119896119894 = (1198841198961198941 119884119896119894119872119896119894)1015840 119884119896 =(1198841198961 119884119896119873119896) and 119884 = (1198841 119884119870) for 119896 = 1 119870 119894 =

1 119873119896 and 119895 = 1 119872119896119894 According to property (P1) andindependent assumption of Brownian motion 119884119896119894 follows amultivariate normal distribution with mean and variance

119896119894 = 120583120578120589119896119879119896119894Σ119896119894 = 1205902120578Σ119896119894 (20)

where

Σ119896119894 = Ω119896119894 + 1205892119896119879119896119894119879119879119896119894119876119896119894 =

[[[[[[[[

1205961198961198941 1205961198961198941 sdot sdot sdot 12059611989611989411205961198961198941 1205961198961198942 sdot sdot sdot 1205961198961198942 d1205961198961198941 1205961198961198942 sdot sdot sdot 120596119896119894119872119896119894

]]]]]]]]

Ω119896119894 = 120589119896119876119896119894 + 2120576119868119896119894

(21)

and 119868119896119894 is an identified matrix of order119872119896119894The log-likelihood function of unknown parameters Θ =120583120578 1205902120578 120573 2120576 is

ln 119871 (Θ | 119884)= minus12 ln (2120587) 119870sum

119896=1

119873119896sum119894=1

119872119894119896 minus 12 ln (1205902120578) 119870sum119896=1

119873119896sum119894=1

119872119896119894minus 12 119870sum119896=1

119873119896sum119894=1

10038161003816100381610038161003816Σ11989611989410038161003816100381610038161003816minus 121205902120578

119870sum119896=1

119873119896sum119894=1

(119910119896119894 minus 120583120578120589119896119879119896119894)1015840 Σminus1119896119894 (119910119896119894 minus 120583120578120589119896119879119896119894)

(22)

6 Mathematical Problems in Engineering

Taking the first partial derivatives of the log-likelihoodfunction with respect to 120583120578 1205902120578 yields

120597 ln 119871 (Θ | 119884)120597120583120578= 11205902120578 (

119870sum119896=1

119873119896sum119894=1

1205891198961198791015840119896119894Σminus1119896119894 119910119896119894 minus 120583120578 119870sum119896=1

119873119896sum119894=1

12058921198961198791015840119896119894Σminus1119896119894 119879119896119894) 120597 ln 119871 (Θ | 119884)1205971205902120578 = minus 121205902120578

119870sum119896=1

119873119896sum119894=1

119872119896119894 + 12 (1205902120578)2sdot 119870sum119896=1

119873119896sum119894=1

(119910119896119894 minus 120583120578120589119896119879119896119894)1015840 Σminus1119896119894 (119910119896119894 minus 120583120578120589119896119879119896119894)

(23)

For the special value of ( 120573 2120576) setting the derivation of120597 ln 119871(Θ | 119884)120597120583120578 120597 ln 119871(Θ | 119884)1205971205902120578 to zero the MLE for120583120578 1205902120578 can be expressed as

120578 = sum119870119896=1sum119873119896119894=1 1205891198961198791015840119896119894Σminus1119896119894 119910119896119894sum119870119896=1sum119873119896119894=1 12058921198961198791015840119896119894Σminus1119896119894 119879119896119894 2120578 = 1sum119870119896=1sum119873119896119894=1119872119896119894

sdot 119870sum119896=1

119873119896sum119894=1

(119910119896119894 minus 120578120589119896119879119896119894)1015840 Σminus1119896119894 (119910119896119894 minus 120578120589119896119879119896119894) (24)

Substituting (24) into (22) and simplifying the profilelog-likelihood function can be written as

ln 119871 (Θ | 119884) = minus12 (ln (2120587) + 1) 119870sum119896=1

119873119896sum119894=1

119872119894119896minus 12 ln (2120578) 119870sum

119896=1

119873119896sum119894=1

119872119896119894minus 12 119870sum119896=1

119873119896sum119894=1

10038161003816100381610038161003816Σ11989611989410038161003816100381610038161003816 (25)

The MLE of 120573 and 2120576 can be obtained by maximizingthe profile log-likelihood function in (25) through a three-dimensional search Then substitute them into (24) we canobtain MLE for 120583120578 and 1205902120578

The value of and 2120576 can be obtained by the followingequations

= 120581 sdot 21205782120576 = 1205902120576 sdot 2120578 (26)

42 Parameter Estimation of SSADT The degradation pro-cess of SSADT shown as (18) is relatively complicated Thuswe introduce the accelerated factor first for the sake ofsimplicity The variable 120577119896 is represented by the product of 1205771and the accelerated factor 1205721198961

120589119896 = 12058911205721198961 (27)

The SSADP can be rewritten as

119909119896119894119895

=

12057811989412058911205961119894119895 + radic1205811205891119861 (12059612119895) 1205911198940 le 120596119894119896119895 le 12059111989411205781198941205891 (1205911119894 + 12057221 (1205962119894119895 minus 1205911119894)) + radic1205811205891119861 (1205911119894 + 12057221 (1205962119894119895 minus 1205911119894)) 1205911119894 le 1205962119894119895 le 1205912119894sdot sdot sdot1205781198941205891(119896minus1sum

119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119896119894119895 minus 120591(119896minus1)119894)) + radic1205811205891119861(119896minus1sum119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119896119894119895 minus 120591(119896minus1)119894)) 120591(119896minus1)119894 le 120596119896119894119895 le 120591119896119894sdot sdot sdot1205781198941205891(119870minus1sum

119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119870119894119895 minus 120591(119870minus1)119894)) + radic1205811205891119861(119870minus1sum119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119870119894119895 minus 120591(119870minus1)119894)) 120591(119870minus1)119894 le 120596119870119894119895 le 120591119870119894

(28)

Specify 120594119894119896119895 = sum119896minus1119899=1 1205721198991(120591119894119899 minus 120591119894(119899minus1)) + 12057221(120596119894119896119895 minus 120591119894(119896minus1))then the SSADP can be expressed a general formula asfollows

119909119896119894119895 = 1205781198941205891120594119896119894119895 + radic1205811205891119861 (120594119896119894119895) 120591(119896minus1)119894 le 120596119896119894119895 ge 120596119896119894119896 = 1 119870 119894 = 1 119873 119895 = 1 119872119896119894(29)

The foregoing transformation is equivalent to convertingthe degradation driven by stress 119878119896 to the degradation understress 1198781 in physics and the converted equivalent time is 120594119896119894119895At the moment the matrix of degradation and equivalenttransformed time are still bidimensional Because of thecontinuity of the degradation process the matrix can bewritten as a column vector with119872119894 element where119872119894 equalsthe sum of119872119894119896 119896 = 1 119870 and the subscript can be written

Mathematical Problems in Engineering 7

as 119898 for the 119894th product 119898 = 1 119872119894 Then the observeddegradation can be expressed as

119910119894119898 = 119909119894119898 + 120576119894119898119909119894119898 = 1205781198941205891120594119894119898 + 1205811205891119861 (120594119894119898) 1 le 119898 le 119870sum

119896=1

119872119894119896(30)

Define Δ1205941198941 = 1205941198941 Δ120594119894119898 = 120594119894119898 minus 120594119894(119898minus1) Δ120594119894 =(Δ1205941198941 Δ120594119894119872119894)1015840 119877119894 = Δ120594119894 and Δ119910119894119898 = 119910119894119898 minus 119910119894(119898minus1) for119894 = 1 119873 119898 = 1 119872119894 Then Δ119910119894 = (Δ1199101198941 Δ119910119894119872119894)1015840follows a multivariate normal distribution 119873(1205831205781205891119877119894 Π119894)where

Π119894 = 119882119894 + 1205902120578120589211198771198941198771015840119894 (119882119901119902)119894 = cov (Δ119910119894119901 Δ119910119894119902)

=

1205811205891119877119894119901 + 1205902120576 119901 = 119902 = 11205811205891119877119894119901 + 21205902120576 119901 = 119902 gt 1minus1205902120576 119901 = 119902 + 1 or 119901 = 119902 minus 10 otherwise

(31)

So the log-likelihood function can be expressed as

ln 119871 (Θ | 119884)= minus12 ln (2120587) 119873sum

119894=1

119872119894 minus 12 119873sum119894=1

ln (1003816100381610038161003816Π1198941003816100381610038161003816)minus 12 119873sum119894=1

(Δ119910119894 minus 1205831205781205891119877119894)1015840Πminus1119894 (Δ119910119894 minus 1205831205781205891119877119894) (32)

where1003816100381610038161003816Π1198941003816100381610038161003816 = 10038161003816100381610038161198821198941003816100381610038161003816 (1 + 1205902120578120589211198771015840119894119882minus1119894 119877119894)Πminus1119894 = 119882minus1119894 minus 1205902120578120589211 + 1205902120578120589211198771015840119894119882minus1119894 119877119894119882minus1119894 1198771198941198771015840119894119882minus1119894 (33)

Taking the first partial derivatives of the log-likelihoodfunction with respect to 120583120578 1205902120578 yields

120597 ln 119871 (Θ | 119884)120597120583120578 = 1205891 119873sum119894=1

1198771015840119894Πminus1119894 Δ119910119894 minus 12058312057812058921 119873sum119894=1

1198771015840119894Πminus1119894 119877119894 (34)

120597 ln 119871 (Θ | 119884)1205971205902120578 = minus 119873sum119894=1

120589211198771015840119894119882minus1119894 1198771 + 1205902120578120589211198771015840119894119882minus1119894 119877119894+ 119873sum119894=1

12058921 (Δ119910119894 minus 1205831205781205891119877119894)1015840119882minus1119894 1198771198941198771015840119894119882minus1119894 (Δ119910119894 minus 1205831205781205891119877119894)(1 + 1205902120578120589211198771015840119894119882minus1119894 119877)2(35)

For the special value of (120581 120573 1205902120576 ) setting the derivation of120597 ln 119871(Θ | 119884)120597120583120578 to zero the restricted MLE for 120583120578 can beexpressed as

120578 = sum119873119894=1 1198771015840119894Πminus1119894 Δ1199101198941205891sum119873119894=1 1198771015840119894Πminus1119894 119877119894 (36)

Similarly set the derivation of 120597 ln 119871(Θ | 119884)120597120583120578 to zeroand the results of the MLE for 1205902120578 will be discussed on twocases

Case 1 It was assumed that the number of the measurementsand the measurement points of each sample are the same forall of the samples under all of the accelerated stressThat is tosay the subscript of 119877119894 Π119894 and119882119894 can be removed

Thus the restricted MLE for 120583120578 can be expressed as

120578 = sum119873119894=1 1198771015840Πminus1119894 Δ11991011989412058911198731198771015840Πminus1119894 119877 (37)

The first partial derivatives of the log-likelihood functionto 1205902120578 can be rewritten as

120597 ln 119871 (Θ | 119884)1205971205902120578 = minus119873 120589211198771015840119882minus11198771 + 1205902120578120589211198771015840119882minus1119877+ 12058921(1 + 1205902120578120589211198771015840119882minus1119877)2sdot 119873sum119894=1

(Δ119910119894 minus 1205831205781205891119877)119882minus11198771198771015840119882minus1 (Δ119910119894 minus 1205831205781205891119877)1015840 (38)

For the special value of (120581 120573 1205902120576 ) setting the derivation of120597 ln 119871(Θ | 119884)1205971205902120578 to zero the result of the MLE for 1205902120578 can beexpressed as2120578

= sum119873119894=1 (Δ119910119894 minus 1205831205781205891119877)119882minus11198771198771015840119882minus1 (Δ119910119894 minus 1205831205781205891119877)101584011987312058921 (1198771015840119882minus1119877)2minus 1120589211198771015840119882minus1119877

(39)

Substituting (37) (39) into (32) and simplifying the profilelog-likelihood function can be written as

ln 119871 (120581 120573 1205902120576 | 120578 2120578 119884) = minus1198721198732 ln (2120587) minus 1198732 minus 1198732sdot ln (|119882|)minus 12

119873sum119894=1

Δ1199101015840119894119882minus1Δ119910119894 minus sum119873119894=1 (1198771015840119882minus1Δ119910119894)21198771015840119882minus1119877 minus 12sdot ln

sum119873119894=1 (1198771015840119882minus1Δ119910119894)21198731198771015840119882minus1119877 minus sum119873119894=1 (1198771015840119882minus1Δ119910119894)211987321198771015840119882minus1119877

(40)

8 Mathematical Problems in Engineering

where119872 is themeasurement time point of each samplewhichis the same for all of the samples based on the assumption ofCase 1

The MLE of 120581 120573 and 1205902120576 can be obtained by maximizingthe profile log-likelihood function in (40) through a three-dimensional searchThen substitute them into (37) and (39)we can obtain the MLE for 120583120578 and 1205902120578 Case 2 The number of the measurements and the measure-ment points of each sample are different for all of the samplesunder all of the accelerated stress In this case the first partialderivatives of the log-likelihood function to 1205902120578 are shown asin (35)There may be no analytical form by setting the partialderivatives to zero The evaluation of 120583120578 can be expressed as(36) Then the profile log-likelihood function of 1205902120578 120581 120573 1205902120576can be written as

ln 119871 (120581 120573 120590120576 1205902120578 | 120578 119884)= minus12 ln (2120587) 119873sum

119894=1

119872119894 minus 12 119873sum119894=1

ln (1003816100381610038161003816Π1198941003816100381610038161003816)minus 119873sum119894=1

(Δ119910119894 minus 1205781205891119877119894)Πminus1119894 (Δ119910119894 minus 1205781205891119877119894)1015840 (41)

The MLE of 1205902120578 120581 120573 and 1205902120576 can be obtained by maxi-mizing the profile log-likelihood function in (41) through afour-dimensional search Then substitute them into (36) wecan obtain the MLE for 120583120578

It is not to say that we can only use degradation forCSADT and increment for SSADT but just make an intro-duction to both of the two methods in the limited space Inaddition we could verify the results by comparing the esti-mation calculated by the two methods to avoid computationerrors

5 Simulation Data Analysis

In order to validate the model described before and theparameter estimation methods simulation test was con-ducted

51 The Simulation Method of CSADT Data The parametersΘ = 120583120578 1205902120578 120581 120573 1205902120576 should be given before the simulationand simulation process is shown as in the following steps

Step 1 Set 119896 = 1Step 2 Generate 119873119894 random numbers with subscripts 119894 119894 =1 2 119873119896 subject to normal distribution with mean 120583120578 andvariance 1205902120578 Step 3 Let 119897 = 0Step 4 Calculate119883119896119894(119897+1) using the Euler approximation [36]

119883119896119894(119897+1) = 119883119896119894119897 + 120578119894120589119896120596 (Δ119905) + 120581120589119896radic120596 (Δ119905)Ψ (42)

where Ψ is the random number according to the standardnormal distribution and Δ119905 is the step size in simulation and1198831198961198940 = 0 As such set 119897 = 119897 + 1 and return to Step 4 to get thedegradation path until 119897 = 119871119896119894+1 where 119871119896119894 is the simulationsteps for 119894th under stress 119878119896Step 5 Set 119896 = 119896 + 1 and return to Step 2 until 119896 = 119870 + 1Step 6 Extract the degradation based on the predefinedmeasurement time point 119905119896119894119895 where 119896 = 1 2 119870 119894 =1 2 119873119896 and 119895 = 1 2 119872119896119894 from the data set 119883119896119894119897 andget the data set of degradation119883119896119894119895Step 7 Simulate the measured degradation 119884119896119894119895 = 119883119896119894119895 + 120576119896119894119895where 120576119896119894119895 sim 119873(0 1205902120576 )52 The Simulation Method of SSADT Data There are somedifferences in the simulation process for SSADT comparedwith CSADT The simulation process is shown as follows

Step 1 Generate 119873 random numbers with subscripts 119894 119894 =1 2 119873 subject to normal distribution with mean 120583120578 andvariance 1205902120578 Step 2 Let 119896 = 1Step 3 Set 119897 = 1Step 4 Calculate119883119896119894(119897+1)

119883119896119894(119897+1) = 119883119896119894119897 + 120578119894120589119896120596 (Δ119905) + 120581120589119896radic120596 (Δ119905)Ψ (43)

where Ψ is the random number according to the standardnormal distribution and Δ119905 is the step size in simulation and1198831198961198940 = 01198831198961198940 = 119883(119896minus1)119894119871(119896minus1) (119896 gt 1) As such set 119897 = 119897+1 andreturn to Step 4 to get the degradation path until 119897 = 119871119896119894 + 1Step 5 Set 119896 = 119896 + 1 and return to Step 2 until 119896 = 119870 + 1Step 6 Extract the degradation based on the predefinedmeasurement time point 119905119896119894119895 where 119896 = 1 2 119870 119894 =1 2 119873 and 119895 = 1 2 119872119894119896 from the data set 119883119896119894119897 andget the data set of degradation119883119896119894119895Step 7 Simulate the measured degradation 119884119896119894119895 = 119883119896119894119895 + 120576119896119894119895where 120576119896119894119895 sim 119873(0 1205902120576 )53 The Analysis of the Simulated SSADT Data We justgive the analysis of simulated SSADT data here becausewe would give a case study of the CSADT later It wasassumed that the accelerated stress is temperature and thesimulation test contains 4 stresseswhich are 50∘C 60∘C 70∘Cand 80∘C For simplicity the transformed time function isset as Λ(119905) = 119905 and 20 degradation paths are generatedThen the parameters for degradation process are assumedas Θ = 5 4 025 minus3000 001 Moreover the time intervalmeasurement is 25 h for each sample of the whole simulationtest with 40 measurement time points The degradation dataare depicted as in Figure 1

Mathematical Problems in Engineering 9

Table 2 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 log-LF AIC RETruth value 5 1 025 mdash minus3000 001 mdash mdash mdash1198720 501 139 028 mdash minus297561 00078 142228 minus283456 084021198721 512 175 mdash 12907 lowast 10minus4 minus298081 00129 112289 minus223577 562711198722 620 mdash mdash 15213 lowast 10minus4 minus305175 13789 lowast 10minus9 109361 minus217721 78049

0 200 400 600 800 1000

0

05

1

15

2

25

th

y

Figure 1 The simulation degradation paths of SSADT

For simplicity the degradation model for SSADT pro-posed in this paper is referred to as1198720 the model presentedby Tang et al [17] as 1198721 and the stochastic SSADT modeldescribed in reference [13] as 1198722 The estimation results ofthe unknown parameters the log-likelihood function valueand the Akaike information criterion (AIC) [37] which isevaluated by (44) are summarized in Table 2 As we cansee from Table 2 our model clearly outperforms model interms of the log-LF and AIC The estimates of 120583120578 1205902120578 and1205902120576 are more close to the truth values compared with 1198721and 1198722 Meanwhile the diffusion-related variables includeparameters 120581 and 120573 and the diffusion parameter under thefour stresses can be obtained by (10) which is (28377 lowast 10minus548534 lowast 10minus5 78244 lowast 10minus5 12001 lowast 10minus4) in 1198721 and(22334lowast10minus5 39912lowast10minus5 64597lowast10minus5 99423lowast10minus5) forthe true model Moreover taking 1198781 as a reference a criterionformulizing the relative error (RE) is shown as (45) It is clearthat1198720 has the smallest RE from Table 2

AIC (119873) = minus2 ln 119871 (Θ) + 2119873 (44)

RE = 1003816100381610038161003816100381610038161003816100381610038161003816120583120578 minus 120578120578

1003816100381610038161003816100381610038161003816100381610038161003816 +100381610038161003816100381610038161003816100381610038161003816100381610038161205902120578 minus 21205782120578

10038161003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161003816(1205902119861)1 minus (2119861)1(2119861)1

10038161003816100381610038161003816100381610038161003816100381610038161003816+ 1003816100381610038161003816100381610038161003816100381610038161003816120573 minus

1003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161205902120576 minus 21205762120576

1003816100381610038161003816100381610038161003816100381610038161003816 (45)

It is assumed that the failure threshold119908 is equal to 5 andthe normal stress is 25∘C The PDF and CDF under normalstress are shown as in Figure 2 The phenomenon can beobserved where the PDF and CDF of our model most nearlyapproached the real model By contrasting 1198721 with 1198722 itcan be concluded that the neglect of random effects wouldresult in sharper PDF which is consistent with the result inreference [17] And it is noteworthy that the 05 quantiles ofthe failure life are almost equal from the CDF of the threemodels But there is a greater difference between the realmodel and 1198721 while the indicator is taken as mean time tofailure as well as the difference between the real model and1198722 So the correlation between the stress and the diffusionparameter should not be neglected Next we are going toprove the superiority of our model in practical application

6 Case Application

The CSADT model with covariates and random effects isverified by the accelerated degradation data of carbon-filmresistors whose raw data set is explicitly given in Table C3of Meeker and Escobar [38]There were 29 samples and threeaccelerated temperature stresses in the whole test where ninesamples were observed at 83∘C the remaining two stresses133∘C and 173∘C had 10 samples for each stress All of thesamples were observed at the same time points with 1199050 = 01199051 = 452 1199052 = 1030 1199053 = 4341 and 1199054 = 8084 (inhours) Similar to [39] it was assumed that the standardoperating temperature was 50∘C and the threshold value forpercent increase in resistance was taken to be 119862 = 12 Itis reasonable and computationally easier to use the ratios inresistance to the initial value for each rather than the percentincrease [9] For ease of calculation the logarithm of the ratiowas taken as the degradation here That is the value 028of the percent increase was changed to 2796119864 minus 3 and soon and the transformed threshold value 119908 was 01133 Thedegradation paths of all the sample were as shown in Figure 3It can be seen that the degradation of the samples uniformlyshowed a nonlinear characteristic especially at the beginningof the ADT Thus the degradation is modeled with the timescale transformedWiener process as (1) with 120596 = Λ(119905) = 119905119887

Similarly the proposed degradation model for SSADT inthis paper is referred to as1198720 themodel presented by Tang etal [17] as1198721 and themodel described in reference [9] as1198723The estimation results of the unknown parameters the log-likelihood function value and the AIC are shown in Table 3The estimated parameters 119887 are approximately equal for thethree models Take 119887 = 050 as an example to show the time

10 Mathematical Problems in Engineering

Table 3 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 b Log-LF AIC MTTF1198720 823 207 108119864 minus 02 mdash minus420282 182119864 minus 20 050 51848 minus102497 380119864 + 071198721 1471 433 mdash 413119864 minus 07 minus458656 506119864 minus 20 053 48637 minus96073 515119864 + 071198722 1111 mdash mdash 676119864 minus 07 minus447982 411119864 minus 18 053 46837 minus92674 406119864 + 07

0 1 2 3 4 50

05

1

15

2

25

th

The P

DF

Real model

times104

times10minus4

M1

M0

M2

0 1 2 3 4 50

02

04

06

08

1

th

The C

DF

times104

Real modelM1

M0

M2

Figure 2 The comparison of the PDF and CDF of the three models for simulated SSADT data

0 2000 4000 6000 80000

0002

0004

0006

0008

001

th

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 2000 4000 6000 80000

001

002

003

th

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 2000 4000 6000 8000th

0

002

004

006

008

01

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 3 The degradation paths of carbon-film resistors

scale transformed degradation paths as in Figure 4 which areapproximately straight lines compared with Figure 3

Our model has the largest log-LF and smallest AICcompared with1198721 and1198722 The result proves that our modelis more suitable for the degradation data Besides 1198721 hasthe larger log-LF and smaller AIC compared with1198722 So therandom effects could not be neglected It can be observed thatthe variance of the drift parameter is slightly largerwhichmaybe owing to the fixed value of diffusion parameter in1198721

The PDF and CDF under the standard operating temper-ature are as shown in Figure 5The time corresponding to thepeak values of PDF were as follows 2676 lowast 107 1049 lowast 107

and 5028lowast106 for1198720 to1198722 But theMTTF of1198720 isminimalfrom Table 3 And the uncertainty in the estimated PDFs ofthe lifetimes under1198720 is smallest compared to1198721 and1198722 asseen in Figure 5When time 119905 approaches positive infinity theCDF of ourmodel which is calculated by (13) tends illimitablyto 1 and the CDF of 1198721 is roughly equal to 09997 It followsthat our model could do well in distinguishing the differencefrom random effects and the effect of covariates Anotherthing which is worth noting is that the measurement error isquite tiny in the case but it is just an individual phenomenonThemeasurement error should also be considered in practicalapplication

Mathematical Problems in Engineering 11

0 20 40 60 800

0002

0004

0006

0008

001

Λ(t)

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 20 40 60 800

001

002

003

Λ(t)

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 20 40 60 800

002

004

006

008

01

Λ(t)

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 4 The degradation paths of carbon-film resistors under transformed time scale

0 2 4 6 8 100

05

1

15

2

25

3

35

th

times10minus8

M0

M1

M2

The P

DF

times107

0 2 4 6 8 100

02

04

06

08

1

th

M0

M1

M2

The C

DF

times107

Figure 5 The comparison of the PDF and CDF of the three models for real CSADT data

7 Conclusion

In this paper the degradation models based on nonlinearWiener process are established for both constant stress accel-erated degradation data and step stress accelerated degrada-tion data Before the establishment the relationship betweenthe drift parameter and stress variables is derived based onthe invariance principle of failure mechanism and Nelsonassumption so is the relationship between the diffusionparameter and stress variables It is concluded that the ratioof drift parameters under two stresses is a constant which isirrelevant to the testing time and depends only on the twostresses as long as the ratio of diffusion parameters is equalto the ratio of drift parameters And the ratio is defined asaccelerated factor Besides the random effects are also takeninto consideration where the drift parameter is assumed to be

normally distributed and the diffusion parameter is same forall of the samples under a certain stress Then the PDF andCDF of the FHT are deduced considering random effects

Because of the dependency between the diffusion param-eter and stress variables the degradation process is quitedifferent either for CSADT or for SSADT The CSADP andSSADP with random effects are modeled Moreover theunknown parameters are solved by MLE based on the twoproperties of Wiener process At the end of the paper thesimulated data of SSADT and the CSADTdata of carbon-filmresistors are both analyzed to verify the proposed model Itis concluded that the model has the biggest log-LF and thesmallest AIC compared with the two other models

The innovation of this paper lies in the following Firstthe random effects are considered under the new relationshipbetween the diffusion parameter and accelerated stresses

12 Mathematical Problems in Engineering

Second the degradation process was modeled for bothCSADT and SSADT Thirdly the unknown parameters wereestimated based on the two properties of Wiener process andthe result of theMLE for 1205902120578 is discussed on two cases Fourththe measurement error of the degradation data is also valued

However we have only considered the random effects ofthe drift parameter in this paper due to the complexity of thecomputation A further research may consider the randomeffects of the diffusion parameter into the model At the sametime the study of the paper may provide new ideas for therelativity analysis between the parameters of other stochasticprocess and stress variables

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by Jiangsu Province GraduateStudent Scientific Research Innovation Project of China(Project KYLX15 0330)The help is gratefully acknowledged

References

[1] H-F Yu ldquoOptimal classification of highly-reliable productswhose degradation paths satisfyWiener processesrdquo EngineeringOptimization vol 35 no 3 pp 313ndash324 2003

[2] XWang N Balakrishnan and B Guo ldquoResidual life estimationbased on a generalized Wiener degradation processrdquo ReliabilityEngineering and System Safety vol 124 pp 13ndash23 2014

[3] Y Ren Q Feng T Ye and B Sun ldquoA novel model of reliabilityassessment for circular electrical connectorsrdquo IEEE Transac-tions on Components Packaging andManufacturing Technologyvol 5 no 6 pp 755ndash761 2015

[4] Z-S Ye and M Xie ldquoStochastic modelling and analysis ofdegradation for highly reliable productsrdquo Applied StochasticModels in Business and Industry vol 31 no 1 pp 16ndash32 2015

[5] H Lim and B-J Yum ldquoOptimal design of accelerated degrada-tion tests based on Wiener process modelsrdquo Journal of AppliedStatistics vol 38 no 2 pp 309ndash325 2011

[6] C J Lu and W Q Meeker ldquoUsing degradation measures toestimate a time-to-failure distributionrdquo Technometrics vol 35no 2 pp 161ndash174 1993

[7] G AWhitmore ldquoEstimating degradation by a wiener diffusionprocess subject to measurement errorrdquo Lifetime Data Analysisvol 1 no 3 pp 307ndash319 1995

[8] Z Pan and N Balakrishnan ldquoReliability modeling of degra-dation of products with multiple performance characteristicsbased on gamma processesrdquo Reliability Engineering amp SystemSafety vol 96 no 8 pp 949ndash957 2011

[9] C Park and W J Padgett ldquoAccelerated degradation modelsfor failure based on geometric Brownian motion and gammaprocessesrdquo Lifetime Data Analysis vol 11 no 4 pp 511ndash5272005

[10] Z-S Ye and N Chen ldquoThe inverse Gaussian process as adegradation modelrdquo Technometrics vol 56 no 3 pp 302ndash3112014

[11] A Onar and W J Padgett ldquoAccelerated test models with theinverse Gaussian distributionrdquo Journal of Statistical Planningand Inference vol 89 no 1-2 pp 119ndash133 2000

[12] T-R Tsai Y L Lio and N Jiang ldquoOptimal decisions on theaccelerated degradation test plan under the Wiener processrdquoQuality Technology and Quantitative Management vol 11 no4 pp 461ndash470 2014

[13] C-M Liao and S-T Tseng ldquoOptimal design for step-stressaccelerated degradation testsrdquo IEEE Transactions on Reliabilityvol 55 no 1 pp 59ndash66 2006

[14] L C Tang G Yang and M Xie ldquoPlanning of step-stress accel-erated degradation testrdquo in Proceedings of the Annual Reliabil-ity and Maintainability SymposiummdashProceedings InternationalSymposium on Product Quality and Integrity pp 287ndash292January 2004

[15] C-H Hu M-Y Lee and J Tang ldquoOptimum step-stress accel-erated degradation test for Wiener degradation process underconstraintsrdquo European Journal of Operational Research vol 241no 2 pp 412ndash421 2015

[16] S Tang ldquoStep stress accelerated degradation process modelingand remaining useful life estimationrdquo Journal of MechanicalEngineering vol 50 no 16 p 33 2014

[17] S Tang X Guo C Yu H Xue and Z Zhou ldquoAccelerated degra-dation tests modeling based on the nonlinear wiener processwith random effectsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 560726 11 pages 2014

[18] L Wang R Pan X Li and T Jiang ldquoA Bayesian reliability eval-uation method with integrated accelerated degradation testingand field informationrdquo Reliability Engineering amp System Safetyvol 112 pp 38ndash47 2013

[19] Z-S Ye N Chen and Y Shen ldquoA new class of Wiener processmodels for degradation analysisrdquo Reliability Engineering ampSystem Safety vol 139 pp 58ndash67 2015

[20] G A Whitmore and F Schenkelberg ldquoModelling accelerateddegradation data using wiener diffusion with a time scaletransformationrdquo Lifetime Data Analysis vol 3 no 1 pp 27ndash451997

[21] K A Doksum and A Hoyland ldquoModels for variable-stressaccelerated life testing experiments based on Wiener processesand the inverse Gaussian distributionrdquo Technometrics vol 34no 1 pp 74ndash82 1992

[22] H Liao and E A Elsayed ldquoReliability inference for field con-ditions from accelerated degradation testingrdquo Naval ResearchLogistics vol 53 no 6 pp 576ndash587 2006

[23] H-WWang T-X Xu andW-YWang ldquoRemaining life predic-tion based on wiener processes with ADT prior informationrdquoQuality and Reliability Engineering International vol 32 no 3pp 753ndash765 2015

[24] C-Y Peng ldquoInverse Gaussian processes with random effectsand explanatory variables for degradation datardquo Technometricsvol 57 no 1 pp 100ndash111 2015

[25] X Pan XHuang Y Chen YWang and R Kang ldquoConnotationof failure mechanism consistency and identificationmethod foraccelerated testingrdquo in Proceedings of the Prognostics and SystemHealth Management Conference (PHM-Shenzhen rsquo11) May 2011

[26] W B Nelson Accelerated Testing Statistical Models Test Plansand Data Analysis John Wiley amp Sons 2009

[27] Z-S Ye Y Hong and Y Xie ldquoHow do heterogeneities in oper-ating environments affect field failure predictions and testplanningrdquo The Annals of Applied Statistics vol 7 no 4 pp2249ndash2271 2013

Mathematical Problems in Engineering 13

[28] Z-X Zhang X-S Si C-H Hu Q Zhang T Li and C XuldquoPlanning repeated degradation testing for products with three-source variabilityrdquo IEEE Transactions on Reliability vol 65 no2 pp 640ndash647 2016

[29] C-Y Peng and S-T Tseng ldquoMis-specification analysis of lineardegradation modelsrdquo IEEE Transactions on Reliability vol 58no 3 pp 444ndash455 2009

[30] X-S Si W Wang C-H Hu D-H Zhou and M G PechtldquoRemaining useful life estimation based on a nonlinear diffu-sion degradation processrdquo IEEE Transactions on Reliability vol61 no 1 pp 50ndash67 2012

[31] X-S Si W Wang C-H Hu M-Y Chen and D-H Zhou ldquoAWiener-process-based degradationmodel with a recursive filteralgorithm for remaining useful life estimationrdquo MechanicalSystems and Signal Processing vol 35 no 1-2 pp 219ndash237 2013

[32] C-C Tsai S-T Tseng and N Balakrishnan ldquoMis-specificationanalyses of gamma andWiener degradation processesrdquo Journalof Statistical Planning and Inference vol 141 no 12 pp 3725ndash3735 2011

[33] F Haghighi and S J Bae ldquoReliability estimation from lineardegradation and failure time data with competing risks under astep-stress accelerated degradation testrdquo IEEE Transactions onReliability vol 64 no 3 pp 960ndash971 2015

[34] Z-S Ye Y Wang K-L Tsui and M Pecht ldquoDegradation dataanalysis usingwiener processeswithmeasurement errorsrdquo IEEETransactions on Reliability vol 62 no 4 pp 772ndash780 2013

[35] J Huang D S Golubovic S Koh et al ldquoDegradation modelingof mid-power white-light LEDs by using Wiener processrdquoOptics Express vol 23 no 15 pp A966ndashA978 2015

[36] A Beskos O Papaspiliopoulos G O Roberts and P Fearn-head ldquoExact and computationally efficient likelihood-basedestimation for discretely observed diffusion processes (withdiscussion)rdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 68 no 3 pp 333ndash382 2006

[37] J Shang and J E Cavanaugh ldquoAn assumption for the develop-ment of bootstrap variants of the Akaike information criterionin mixed modelsrdquo Statistics and Probability Letters vol 78 no12 pp 1422ndash1429 2008

[38] W QMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[39] W J Padgett andM A Tomlinson ldquoInference from accelerateddegradation and failure data based on Gaussian process mod-elsrdquo Lifetime Data Analysis vol 10 no 2 pp 191ndash206 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

According to the analysis the degradation process ofSSADT can be formulated as

119909119896119894119895 =

1205781198941205891120596119894119896119895 + radic1205811205891119861 (120596119894119896119895) 1205911198940 le 120596119894119896119895 le 120591119894112057811989412058911205911198941 + radic1205811205891119861 (1205911198941) + 1205781198941205892 (120596119894119896119895 minus 1205911198941) + radic1205811205892119861 (120596119894119896119895 minus 1205911198941) 1205911198941 le 120596119894119896119895 le 1205911198942sdot sdot sdot119896minus1sum119899=1

[120578119894120589119899 (120591119894119899 minus 120591119894(119899minus1)) + radic120581120589119899119861 (120591119894119899 minus 120591119894(119899minus1))] + 120578119894120589119896 (120596119894119896119895 minus 1205911198942) + radic120581120589119896119861 (120596119894119896119895 minus 1205911198942) 120591119894119896minus1 le 120596119894119896119895 le 120591119894119896sdot sdot sdot119870minus1sum119899=1

[120578119894120589119899 (120591119894119899 minus 120591119894(119899minus1)) + 120581120589119899119861 (120591119894119899 minus 120591119894(119899minus1))] + 120578119894120589119870 (120596119894119896119895 minus 1205911198942) + 120581120589119870119861 (120596119894119896119895 minus 1205911198942) 120591119894119870minus1 le 120596119894119896119895 le 120591119894119870

(18)

4 Parameter Estimation

In real applications it is inevitable that some measurementerrors may be introduced during the observation process[34] When a measurement is taken the observed degrada-tion is shown as follows for both CSADT and SSADT

119910119896119894119895 (120596119896119894119895 | 119878119896) = 119909119896119894119895 (120596119896119894119895 | 119878119896) + 120576119896119894119895119896 = 1 119870 119894 = 1 119873119896 119895 = 1 119872119896119894 (19)

where the measurement errors 120576119896119894119895 are assumed to be iidrealizations of 120576119896119894119895 sim 119873(0 1205902120576 ) and mutually independent of119909119896119894119895 [7 34] and 119873119896 is equal to 119873 for all of the acceleratedstresses of SSADT

The unknown parameters in the models are Θ =120583120578 1205902120578 120581 120573 1205902120576 There are two methods to deal with theunknown parameters with each considering one of thetwo main properties of Brownian motion which could becharacterized as follows [35]

(P1) The increment Δ119909(120596) = 119909(120596 + Δ120596) minus 119909(120596) isindependent of the degradation 119909(120596) which meansthat if 0 le 1205961 le 1205962 le 1205963 le 1205964 then 119909(1205962) minus119909(1205961) and 119909(1205964) minus 119909(1205963) are independent randomvariables Meanwhile the increment Δ119909(120596) follows anormal distribution where the mean equals 120582Δ120596 andthe variance is 1205902119861Δ120596

(P2) According to (P1) the degradation119909(120596) follows a nor-mal distribution with the mean 120582120596 and the variance1205902119861120596

Even though our models concern the unit-to-unitevariability the essence of the Wiener process remains thesame Owing to space constraints this paper deals with theunknown parameter based on property (P2) in the case ofCSADT and property (P1) in the case of SSADT

41 Parameter Estimation of CSADT Specify 119879119896119894 = (1198791198961198941 119879119896119894119872119896119894)1015840 119879119896119894119895 = 120596119896119894119895 119884119896119894 = (1198841198961198941 119884119896119894119872119896119894)1015840 119884119896 =(1198841198961 119884119896119873119896) and 119884 = (1198841 119884119870) for 119896 = 1 119870 119894 =

1 119873119896 and 119895 = 1 119872119896119894 According to property (P1) andindependent assumption of Brownian motion 119884119896119894 follows amultivariate normal distribution with mean and variance

119896119894 = 120583120578120589119896119879119896119894Σ119896119894 = 1205902120578Σ119896119894 (20)

where

Σ119896119894 = Ω119896119894 + 1205892119896119879119896119894119879119879119896119894119876119896119894 =

[[[[[[[[

1205961198961198941 1205961198961198941 sdot sdot sdot 12059611989611989411205961198961198941 1205961198961198942 sdot sdot sdot 1205961198961198942 d1205961198961198941 1205961198961198942 sdot sdot sdot 120596119896119894119872119896119894

]]]]]]]]

Ω119896119894 = 120589119896119876119896119894 + 2120576119868119896119894

(21)

and 119868119896119894 is an identified matrix of order119872119896119894The log-likelihood function of unknown parameters Θ =120583120578 1205902120578 120573 2120576 is

ln 119871 (Θ | 119884)= minus12 ln (2120587) 119870sum

119896=1

119873119896sum119894=1

119872119894119896 minus 12 ln (1205902120578) 119870sum119896=1

119873119896sum119894=1

119872119896119894minus 12 119870sum119896=1

119873119896sum119894=1

10038161003816100381610038161003816Σ11989611989410038161003816100381610038161003816minus 121205902120578

119870sum119896=1

119873119896sum119894=1

(119910119896119894 minus 120583120578120589119896119879119896119894)1015840 Σminus1119896119894 (119910119896119894 minus 120583120578120589119896119879119896119894)

(22)

6 Mathematical Problems in Engineering

Taking the first partial derivatives of the log-likelihoodfunction with respect to 120583120578 1205902120578 yields

120597 ln 119871 (Θ | 119884)120597120583120578= 11205902120578 (

119870sum119896=1

119873119896sum119894=1

1205891198961198791015840119896119894Σminus1119896119894 119910119896119894 minus 120583120578 119870sum119896=1

119873119896sum119894=1

12058921198961198791015840119896119894Σminus1119896119894 119879119896119894) 120597 ln 119871 (Θ | 119884)1205971205902120578 = minus 121205902120578

119870sum119896=1

119873119896sum119894=1

119872119896119894 + 12 (1205902120578)2sdot 119870sum119896=1

119873119896sum119894=1

(119910119896119894 minus 120583120578120589119896119879119896119894)1015840 Σminus1119896119894 (119910119896119894 minus 120583120578120589119896119879119896119894)

(23)

For the special value of ( 120573 2120576) setting the derivation of120597 ln 119871(Θ | 119884)120597120583120578 120597 ln 119871(Θ | 119884)1205971205902120578 to zero the MLE for120583120578 1205902120578 can be expressed as

120578 = sum119870119896=1sum119873119896119894=1 1205891198961198791015840119896119894Σminus1119896119894 119910119896119894sum119870119896=1sum119873119896119894=1 12058921198961198791015840119896119894Σminus1119896119894 119879119896119894 2120578 = 1sum119870119896=1sum119873119896119894=1119872119896119894

sdot 119870sum119896=1

119873119896sum119894=1

(119910119896119894 minus 120578120589119896119879119896119894)1015840 Σminus1119896119894 (119910119896119894 minus 120578120589119896119879119896119894) (24)

Substituting (24) into (22) and simplifying the profilelog-likelihood function can be written as

ln 119871 (Θ | 119884) = minus12 (ln (2120587) + 1) 119870sum119896=1

119873119896sum119894=1

119872119894119896minus 12 ln (2120578) 119870sum

119896=1

119873119896sum119894=1

119872119896119894minus 12 119870sum119896=1

119873119896sum119894=1

10038161003816100381610038161003816Σ11989611989410038161003816100381610038161003816 (25)

The MLE of 120573 and 2120576 can be obtained by maximizingthe profile log-likelihood function in (25) through a three-dimensional search Then substitute them into (24) we canobtain MLE for 120583120578 and 1205902120578

The value of and 2120576 can be obtained by the followingequations

= 120581 sdot 21205782120576 = 1205902120576 sdot 2120578 (26)

42 Parameter Estimation of SSADT The degradation pro-cess of SSADT shown as (18) is relatively complicated Thuswe introduce the accelerated factor first for the sake ofsimplicity The variable 120577119896 is represented by the product of 1205771and the accelerated factor 1205721198961

120589119896 = 12058911205721198961 (27)

The SSADP can be rewritten as

119909119896119894119895

=

12057811989412058911205961119894119895 + radic1205811205891119861 (12059612119895) 1205911198940 le 120596119894119896119895 le 12059111989411205781198941205891 (1205911119894 + 12057221 (1205962119894119895 minus 1205911119894)) + radic1205811205891119861 (1205911119894 + 12057221 (1205962119894119895 minus 1205911119894)) 1205911119894 le 1205962119894119895 le 1205912119894sdot sdot sdot1205781198941205891(119896minus1sum

119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119896119894119895 minus 120591(119896minus1)119894)) + radic1205811205891119861(119896minus1sum119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119896119894119895 minus 120591(119896minus1)119894)) 120591(119896minus1)119894 le 120596119896119894119895 le 120591119896119894sdot sdot sdot1205781198941205891(119870minus1sum

119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119870119894119895 minus 120591(119870minus1)119894)) + radic1205811205891119861(119870minus1sum119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119870119894119895 minus 120591(119870minus1)119894)) 120591(119870minus1)119894 le 120596119870119894119895 le 120591119870119894

(28)

Specify 120594119894119896119895 = sum119896minus1119899=1 1205721198991(120591119894119899 minus 120591119894(119899minus1)) + 12057221(120596119894119896119895 minus 120591119894(119896minus1))then the SSADP can be expressed a general formula asfollows

119909119896119894119895 = 1205781198941205891120594119896119894119895 + radic1205811205891119861 (120594119896119894119895) 120591(119896minus1)119894 le 120596119896119894119895 ge 120596119896119894119896 = 1 119870 119894 = 1 119873 119895 = 1 119872119896119894(29)

The foregoing transformation is equivalent to convertingthe degradation driven by stress 119878119896 to the degradation understress 1198781 in physics and the converted equivalent time is 120594119896119894119895At the moment the matrix of degradation and equivalenttransformed time are still bidimensional Because of thecontinuity of the degradation process the matrix can bewritten as a column vector with119872119894 element where119872119894 equalsthe sum of119872119894119896 119896 = 1 119870 and the subscript can be written

Mathematical Problems in Engineering 7

as 119898 for the 119894th product 119898 = 1 119872119894 Then the observeddegradation can be expressed as

119910119894119898 = 119909119894119898 + 120576119894119898119909119894119898 = 1205781198941205891120594119894119898 + 1205811205891119861 (120594119894119898) 1 le 119898 le 119870sum

119896=1

119872119894119896(30)

Define Δ1205941198941 = 1205941198941 Δ120594119894119898 = 120594119894119898 minus 120594119894(119898minus1) Δ120594119894 =(Δ1205941198941 Δ120594119894119872119894)1015840 119877119894 = Δ120594119894 and Δ119910119894119898 = 119910119894119898 minus 119910119894(119898minus1) for119894 = 1 119873 119898 = 1 119872119894 Then Δ119910119894 = (Δ1199101198941 Δ119910119894119872119894)1015840follows a multivariate normal distribution 119873(1205831205781205891119877119894 Π119894)where

Π119894 = 119882119894 + 1205902120578120589211198771198941198771015840119894 (119882119901119902)119894 = cov (Δ119910119894119901 Δ119910119894119902)

=

1205811205891119877119894119901 + 1205902120576 119901 = 119902 = 11205811205891119877119894119901 + 21205902120576 119901 = 119902 gt 1minus1205902120576 119901 = 119902 + 1 or 119901 = 119902 minus 10 otherwise

(31)

So the log-likelihood function can be expressed as

ln 119871 (Θ | 119884)= minus12 ln (2120587) 119873sum

119894=1

119872119894 minus 12 119873sum119894=1

ln (1003816100381610038161003816Π1198941003816100381610038161003816)minus 12 119873sum119894=1

(Δ119910119894 minus 1205831205781205891119877119894)1015840Πminus1119894 (Δ119910119894 minus 1205831205781205891119877119894) (32)

where1003816100381610038161003816Π1198941003816100381610038161003816 = 10038161003816100381610038161198821198941003816100381610038161003816 (1 + 1205902120578120589211198771015840119894119882minus1119894 119877119894)Πminus1119894 = 119882minus1119894 minus 1205902120578120589211 + 1205902120578120589211198771015840119894119882minus1119894 119877119894119882minus1119894 1198771198941198771015840119894119882minus1119894 (33)

Taking the first partial derivatives of the log-likelihoodfunction with respect to 120583120578 1205902120578 yields

120597 ln 119871 (Θ | 119884)120597120583120578 = 1205891 119873sum119894=1

1198771015840119894Πminus1119894 Δ119910119894 minus 12058312057812058921 119873sum119894=1

1198771015840119894Πminus1119894 119877119894 (34)

120597 ln 119871 (Θ | 119884)1205971205902120578 = minus 119873sum119894=1

120589211198771015840119894119882minus1119894 1198771 + 1205902120578120589211198771015840119894119882minus1119894 119877119894+ 119873sum119894=1

12058921 (Δ119910119894 minus 1205831205781205891119877119894)1015840119882minus1119894 1198771198941198771015840119894119882minus1119894 (Δ119910119894 minus 1205831205781205891119877119894)(1 + 1205902120578120589211198771015840119894119882minus1119894 119877)2(35)

For the special value of (120581 120573 1205902120576 ) setting the derivation of120597 ln 119871(Θ | 119884)120597120583120578 to zero the restricted MLE for 120583120578 can beexpressed as

120578 = sum119873119894=1 1198771015840119894Πminus1119894 Δ1199101198941205891sum119873119894=1 1198771015840119894Πminus1119894 119877119894 (36)

Similarly set the derivation of 120597 ln 119871(Θ | 119884)120597120583120578 to zeroand the results of the MLE for 1205902120578 will be discussed on twocases

Case 1 It was assumed that the number of the measurementsand the measurement points of each sample are the same forall of the samples under all of the accelerated stressThat is tosay the subscript of 119877119894 Π119894 and119882119894 can be removed

Thus the restricted MLE for 120583120578 can be expressed as

120578 = sum119873119894=1 1198771015840Πminus1119894 Δ11991011989412058911198731198771015840Πminus1119894 119877 (37)

The first partial derivatives of the log-likelihood functionto 1205902120578 can be rewritten as

120597 ln 119871 (Θ | 119884)1205971205902120578 = minus119873 120589211198771015840119882minus11198771 + 1205902120578120589211198771015840119882minus1119877+ 12058921(1 + 1205902120578120589211198771015840119882minus1119877)2sdot 119873sum119894=1

(Δ119910119894 minus 1205831205781205891119877)119882minus11198771198771015840119882minus1 (Δ119910119894 minus 1205831205781205891119877)1015840 (38)

For the special value of (120581 120573 1205902120576 ) setting the derivation of120597 ln 119871(Θ | 119884)1205971205902120578 to zero the result of the MLE for 1205902120578 can beexpressed as2120578

= sum119873119894=1 (Δ119910119894 minus 1205831205781205891119877)119882minus11198771198771015840119882minus1 (Δ119910119894 minus 1205831205781205891119877)101584011987312058921 (1198771015840119882minus1119877)2minus 1120589211198771015840119882minus1119877

(39)

Substituting (37) (39) into (32) and simplifying the profilelog-likelihood function can be written as

ln 119871 (120581 120573 1205902120576 | 120578 2120578 119884) = minus1198721198732 ln (2120587) minus 1198732 minus 1198732sdot ln (|119882|)minus 12

119873sum119894=1

Δ1199101015840119894119882minus1Δ119910119894 minus sum119873119894=1 (1198771015840119882minus1Δ119910119894)21198771015840119882minus1119877 minus 12sdot ln

sum119873119894=1 (1198771015840119882minus1Δ119910119894)21198731198771015840119882minus1119877 minus sum119873119894=1 (1198771015840119882minus1Δ119910119894)211987321198771015840119882minus1119877

(40)

8 Mathematical Problems in Engineering

where119872 is themeasurement time point of each samplewhichis the same for all of the samples based on the assumption ofCase 1

The MLE of 120581 120573 and 1205902120576 can be obtained by maximizingthe profile log-likelihood function in (40) through a three-dimensional searchThen substitute them into (37) and (39)we can obtain the MLE for 120583120578 and 1205902120578 Case 2 The number of the measurements and the measure-ment points of each sample are different for all of the samplesunder all of the accelerated stress In this case the first partialderivatives of the log-likelihood function to 1205902120578 are shown asin (35)There may be no analytical form by setting the partialderivatives to zero The evaluation of 120583120578 can be expressed as(36) Then the profile log-likelihood function of 1205902120578 120581 120573 1205902120576can be written as

ln 119871 (120581 120573 120590120576 1205902120578 | 120578 119884)= minus12 ln (2120587) 119873sum

119894=1

119872119894 minus 12 119873sum119894=1

ln (1003816100381610038161003816Π1198941003816100381610038161003816)minus 119873sum119894=1

(Δ119910119894 minus 1205781205891119877119894)Πminus1119894 (Δ119910119894 minus 1205781205891119877119894)1015840 (41)

The MLE of 1205902120578 120581 120573 and 1205902120576 can be obtained by maxi-mizing the profile log-likelihood function in (41) through afour-dimensional search Then substitute them into (36) wecan obtain the MLE for 120583120578

It is not to say that we can only use degradation forCSADT and increment for SSADT but just make an intro-duction to both of the two methods in the limited space Inaddition we could verify the results by comparing the esti-mation calculated by the two methods to avoid computationerrors

5 Simulation Data Analysis

In order to validate the model described before and theparameter estimation methods simulation test was con-ducted

51 The Simulation Method of CSADT Data The parametersΘ = 120583120578 1205902120578 120581 120573 1205902120576 should be given before the simulationand simulation process is shown as in the following steps

Step 1 Set 119896 = 1Step 2 Generate 119873119894 random numbers with subscripts 119894 119894 =1 2 119873119896 subject to normal distribution with mean 120583120578 andvariance 1205902120578 Step 3 Let 119897 = 0Step 4 Calculate119883119896119894(119897+1) using the Euler approximation [36]

119883119896119894(119897+1) = 119883119896119894119897 + 120578119894120589119896120596 (Δ119905) + 120581120589119896radic120596 (Δ119905)Ψ (42)

where Ψ is the random number according to the standardnormal distribution and Δ119905 is the step size in simulation and1198831198961198940 = 0 As such set 119897 = 119897 + 1 and return to Step 4 to get thedegradation path until 119897 = 119871119896119894+1 where 119871119896119894 is the simulationsteps for 119894th under stress 119878119896Step 5 Set 119896 = 119896 + 1 and return to Step 2 until 119896 = 119870 + 1Step 6 Extract the degradation based on the predefinedmeasurement time point 119905119896119894119895 where 119896 = 1 2 119870 119894 =1 2 119873119896 and 119895 = 1 2 119872119896119894 from the data set 119883119896119894119897 andget the data set of degradation119883119896119894119895Step 7 Simulate the measured degradation 119884119896119894119895 = 119883119896119894119895 + 120576119896119894119895where 120576119896119894119895 sim 119873(0 1205902120576 )52 The Simulation Method of SSADT Data There are somedifferences in the simulation process for SSADT comparedwith CSADT The simulation process is shown as follows

Step 1 Generate 119873 random numbers with subscripts 119894 119894 =1 2 119873 subject to normal distribution with mean 120583120578 andvariance 1205902120578 Step 2 Let 119896 = 1Step 3 Set 119897 = 1Step 4 Calculate119883119896119894(119897+1)

119883119896119894(119897+1) = 119883119896119894119897 + 120578119894120589119896120596 (Δ119905) + 120581120589119896radic120596 (Δ119905)Ψ (43)

where Ψ is the random number according to the standardnormal distribution and Δ119905 is the step size in simulation and1198831198961198940 = 01198831198961198940 = 119883(119896minus1)119894119871(119896minus1) (119896 gt 1) As such set 119897 = 119897+1 andreturn to Step 4 to get the degradation path until 119897 = 119871119896119894 + 1Step 5 Set 119896 = 119896 + 1 and return to Step 2 until 119896 = 119870 + 1Step 6 Extract the degradation based on the predefinedmeasurement time point 119905119896119894119895 where 119896 = 1 2 119870 119894 =1 2 119873 and 119895 = 1 2 119872119894119896 from the data set 119883119896119894119897 andget the data set of degradation119883119896119894119895Step 7 Simulate the measured degradation 119884119896119894119895 = 119883119896119894119895 + 120576119896119894119895where 120576119896119894119895 sim 119873(0 1205902120576 )53 The Analysis of the Simulated SSADT Data We justgive the analysis of simulated SSADT data here becausewe would give a case study of the CSADT later It wasassumed that the accelerated stress is temperature and thesimulation test contains 4 stresseswhich are 50∘C 60∘C 70∘Cand 80∘C For simplicity the transformed time function isset as Λ(119905) = 119905 and 20 degradation paths are generatedThen the parameters for degradation process are assumedas Θ = 5 4 025 minus3000 001 Moreover the time intervalmeasurement is 25 h for each sample of the whole simulationtest with 40 measurement time points The degradation dataare depicted as in Figure 1

Mathematical Problems in Engineering 9

Table 2 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 log-LF AIC RETruth value 5 1 025 mdash minus3000 001 mdash mdash mdash1198720 501 139 028 mdash minus297561 00078 142228 minus283456 084021198721 512 175 mdash 12907 lowast 10minus4 minus298081 00129 112289 minus223577 562711198722 620 mdash mdash 15213 lowast 10minus4 minus305175 13789 lowast 10minus9 109361 minus217721 78049

0 200 400 600 800 1000

0

05

1

15

2

25

th

y

Figure 1 The simulation degradation paths of SSADT

For simplicity the degradation model for SSADT pro-posed in this paper is referred to as1198720 the model presentedby Tang et al [17] as 1198721 and the stochastic SSADT modeldescribed in reference [13] as 1198722 The estimation results ofthe unknown parameters the log-likelihood function valueand the Akaike information criterion (AIC) [37] which isevaluated by (44) are summarized in Table 2 As we cansee from Table 2 our model clearly outperforms model interms of the log-LF and AIC The estimates of 120583120578 1205902120578 and1205902120576 are more close to the truth values compared with 1198721and 1198722 Meanwhile the diffusion-related variables includeparameters 120581 and 120573 and the diffusion parameter under thefour stresses can be obtained by (10) which is (28377 lowast 10minus548534 lowast 10minus5 78244 lowast 10minus5 12001 lowast 10minus4) in 1198721 and(22334lowast10minus5 39912lowast10minus5 64597lowast10minus5 99423lowast10minus5) forthe true model Moreover taking 1198781 as a reference a criterionformulizing the relative error (RE) is shown as (45) It is clearthat1198720 has the smallest RE from Table 2

AIC (119873) = minus2 ln 119871 (Θ) + 2119873 (44)

RE = 1003816100381610038161003816100381610038161003816100381610038161003816120583120578 minus 120578120578

1003816100381610038161003816100381610038161003816100381610038161003816 +100381610038161003816100381610038161003816100381610038161003816100381610038161205902120578 minus 21205782120578

10038161003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161003816(1205902119861)1 minus (2119861)1(2119861)1

10038161003816100381610038161003816100381610038161003816100381610038161003816+ 1003816100381610038161003816100381610038161003816100381610038161003816120573 minus

1003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161205902120576 minus 21205762120576

1003816100381610038161003816100381610038161003816100381610038161003816 (45)

It is assumed that the failure threshold119908 is equal to 5 andthe normal stress is 25∘C The PDF and CDF under normalstress are shown as in Figure 2 The phenomenon can beobserved where the PDF and CDF of our model most nearlyapproached the real model By contrasting 1198721 with 1198722 itcan be concluded that the neglect of random effects wouldresult in sharper PDF which is consistent with the result inreference [17] And it is noteworthy that the 05 quantiles ofthe failure life are almost equal from the CDF of the threemodels But there is a greater difference between the realmodel and 1198721 while the indicator is taken as mean time tofailure as well as the difference between the real model and1198722 So the correlation between the stress and the diffusionparameter should not be neglected Next we are going toprove the superiority of our model in practical application

6 Case Application

The CSADT model with covariates and random effects isverified by the accelerated degradation data of carbon-filmresistors whose raw data set is explicitly given in Table C3of Meeker and Escobar [38]There were 29 samples and threeaccelerated temperature stresses in the whole test where ninesamples were observed at 83∘C the remaining two stresses133∘C and 173∘C had 10 samples for each stress All of thesamples were observed at the same time points with 1199050 = 01199051 = 452 1199052 = 1030 1199053 = 4341 and 1199054 = 8084 (inhours) Similar to [39] it was assumed that the standardoperating temperature was 50∘C and the threshold value forpercent increase in resistance was taken to be 119862 = 12 Itis reasonable and computationally easier to use the ratios inresistance to the initial value for each rather than the percentincrease [9] For ease of calculation the logarithm of the ratiowas taken as the degradation here That is the value 028of the percent increase was changed to 2796119864 minus 3 and soon and the transformed threshold value 119908 was 01133 Thedegradation paths of all the sample were as shown in Figure 3It can be seen that the degradation of the samples uniformlyshowed a nonlinear characteristic especially at the beginningof the ADT Thus the degradation is modeled with the timescale transformedWiener process as (1) with 120596 = Λ(119905) = 119905119887

Similarly the proposed degradation model for SSADT inthis paper is referred to as1198720 themodel presented by Tang etal [17] as1198721 and themodel described in reference [9] as1198723The estimation results of the unknown parameters the log-likelihood function value and the AIC are shown in Table 3The estimated parameters 119887 are approximately equal for thethree models Take 119887 = 050 as an example to show the time

10 Mathematical Problems in Engineering

Table 3 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 b Log-LF AIC MTTF1198720 823 207 108119864 minus 02 mdash minus420282 182119864 minus 20 050 51848 minus102497 380119864 + 071198721 1471 433 mdash 413119864 minus 07 minus458656 506119864 minus 20 053 48637 minus96073 515119864 + 071198722 1111 mdash mdash 676119864 minus 07 minus447982 411119864 minus 18 053 46837 minus92674 406119864 + 07

0 1 2 3 4 50

05

1

15

2

25

th

The P

DF

Real model

times104

times10minus4

M1

M0

M2

0 1 2 3 4 50

02

04

06

08

1

th

The C

DF

times104

Real modelM1

M0

M2

Figure 2 The comparison of the PDF and CDF of the three models for simulated SSADT data

0 2000 4000 6000 80000

0002

0004

0006

0008

001

th

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 2000 4000 6000 80000

001

002

003

th

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 2000 4000 6000 8000th

0

002

004

006

008

01

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 3 The degradation paths of carbon-film resistors

scale transformed degradation paths as in Figure 4 which areapproximately straight lines compared with Figure 3

Our model has the largest log-LF and smallest AICcompared with1198721 and1198722 The result proves that our modelis more suitable for the degradation data Besides 1198721 hasthe larger log-LF and smaller AIC compared with1198722 So therandom effects could not be neglected It can be observed thatthe variance of the drift parameter is slightly largerwhichmaybe owing to the fixed value of diffusion parameter in1198721

The PDF and CDF under the standard operating temper-ature are as shown in Figure 5The time corresponding to thepeak values of PDF were as follows 2676 lowast 107 1049 lowast 107

and 5028lowast106 for1198720 to1198722 But theMTTF of1198720 isminimalfrom Table 3 And the uncertainty in the estimated PDFs ofthe lifetimes under1198720 is smallest compared to1198721 and1198722 asseen in Figure 5When time 119905 approaches positive infinity theCDF of ourmodel which is calculated by (13) tends illimitablyto 1 and the CDF of 1198721 is roughly equal to 09997 It followsthat our model could do well in distinguishing the differencefrom random effects and the effect of covariates Anotherthing which is worth noting is that the measurement error isquite tiny in the case but it is just an individual phenomenonThemeasurement error should also be considered in practicalapplication

Mathematical Problems in Engineering 11

0 20 40 60 800

0002

0004

0006

0008

001

Λ(t)

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 20 40 60 800

001

002

003

Λ(t)

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 20 40 60 800

002

004

006

008

01

Λ(t)

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 4 The degradation paths of carbon-film resistors under transformed time scale

0 2 4 6 8 100

05

1

15

2

25

3

35

th

times10minus8

M0

M1

M2

The P

DF

times107

0 2 4 6 8 100

02

04

06

08

1

th

M0

M1

M2

The C

DF

times107

Figure 5 The comparison of the PDF and CDF of the three models for real CSADT data

7 Conclusion

In this paper the degradation models based on nonlinearWiener process are established for both constant stress accel-erated degradation data and step stress accelerated degrada-tion data Before the establishment the relationship betweenthe drift parameter and stress variables is derived based onthe invariance principle of failure mechanism and Nelsonassumption so is the relationship between the diffusionparameter and stress variables It is concluded that the ratioof drift parameters under two stresses is a constant which isirrelevant to the testing time and depends only on the twostresses as long as the ratio of diffusion parameters is equalto the ratio of drift parameters And the ratio is defined asaccelerated factor Besides the random effects are also takeninto consideration where the drift parameter is assumed to be

normally distributed and the diffusion parameter is same forall of the samples under a certain stress Then the PDF andCDF of the FHT are deduced considering random effects

Because of the dependency between the diffusion param-eter and stress variables the degradation process is quitedifferent either for CSADT or for SSADT The CSADP andSSADP with random effects are modeled Moreover theunknown parameters are solved by MLE based on the twoproperties of Wiener process At the end of the paper thesimulated data of SSADT and the CSADTdata of carbon-filmresistors are both analyzed to verify the proposed model Itis concluded that the model has the biggest log-LF and thesmallest AIC compared with the two other models

The innovation of this paper lies in the following Firstthe random effects are considered under the new relationshipbetween the diffusion parameter and accelerated stresses

12 Mathematical Problems in Engineering

Second the degradation process was modeled for bothCSADT and SSADT Thirdly the unknown parameters wereestimated based on the two properties of Wiener process andthe result of theMLE for 1205902120578 is discussed on two cases Fourththe measurement error of the degradation data is also valued

However we have only considered the random effects ofthe drift parameter in this paper due to the complexity of thecomputation A further research may consider the randomeffects of the diffusion parameter into the model At the sametime the study of the paper may provide new ideas for therelativity analysis between the parameters of other stochasticprocess and stress variables

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by Jiangsu Province GraduateStudent Scientific Research Innovation Project of China(Project KYLX15 0330)The help is gratefully acknowledged

References

[1] H-F Yu ldquoOptimal classification of highly-reliable productswhose degradation paths satisfyWiener processesrdquo EngineeringOptimization vol 35 no 3 pp 313ndash324 2003

[2] XWang N Balakrishnan and B Guo ldquoResidual life estimationbased on a generalized Wiener degradation processrdquo ReliabilityEngineering and System Safety vol 124 pp 13ndash23 2014

[3] Y Ren Q Feng T Ye and B Sun ldquoA novel model of reliabilityassessment for circular electrical connectorsrdquo IEEE Transac-tions on Components Packaging andManufacturing Technologyvol 5 no 6 pp 755ndash761 2015

[4] Z-S Ye and M Xie ldquoStochastic modelling and analysis ofdegradation for highly reliable productsrdquo Applied StochasticModels in Business and Industry vol 31 no 1 pp 16ndash32 2015

[5] H Lim and B-J Yum ldquoOptimal design of accelerated degrada-tion tests based on Wiener process modelsrdquo Journal of AppliedStatistics vol 38 no 2 pp 309ndash325 2011

[6] C J Lu and W Q Meeker ldquoUsing degradation measures toestimate a time-to-failure distributionrdquo Technometrics vol 35no 2 pp 161ndash174 1993

[7] G AWhitmore ldquoEstimating degradation by a wiener diffusionprocess subject to measurement errorrdquo Lifetime Data Analysisvol 1 no 3 pp 307ndash319 1995

[8] Z Pan and N Balakrishnan ldquoReliability modeling of degra-dation of products with multiple performance characteristicsbased on gamma processesrdquo Reliability Engineering amp SystemSafety vol 96 no 8 pp 949ndash957 2011

[9] C Park and W J Padgett ldquoAccelerated degradation modelsfor failure based on geometric Brownian motion and gammaprocessesrdquo Lifetime Data Analysis vol 11 no 4 pp 511ndash5272005

[10] Z-S Ye and N Chen ldquoThe inverse Gaussian process as adegradation modelrdquo Technometrics vol 56 no 3 pp 302ndash3112014

[11] A Onar and W J Padgett ldquoAccelerated test models with theinverse Gaussian distributionrdquo Journal of Statistical Planningand Inference vol 89 no 1-2 pp 119ndash133 2000

[12] T-R Tsai Y L Lio and N Jiang ldquoOptimal decisions on theaccelerated degradation test plan under the Wiener processrdquoQuality Technology and Quantitative Management vol 11 no4 pp 461ndash470 2014

[13] C-M Liao and S-T Tseng ldquoOptimal design for step-stressaccelerated degradation testsrdquo IEEE Transactions on Reliabilityvol 55 no 1 pp 59ndash66 2006

[14] L C Tang G Yang and M Xie ldquoPlanning of step-stress accel-erated degradation testrdquo in Proceedings of the Annual Reliabil-ity and Maintainability SymposiummdashProceedings InternationalSymposium on Product Quality and Integrity pp 287ndash292January 2004

[15] C-H Hu M-Y Lee and J Tang ldquoOptimum step-stress accel-erated degradation test for Wiener degradation process underconstraintsrdquo European Journal of Operational Research vol 241no 2 pp 412ndash421 2015

[16] S Tang ldquoStep stress accelerated degradation process modelingand remaining useful life estimationrdquo Journal of MechanicalEngineering vol 50 no 16 p 33 2014

[17] S Tang X Guo C Yu H Xue and Z Zhou ldquoAccelerated degra-dation tests modeling based on the nonlinear wiener processwith random effectsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 560726 11 pages 2014

[18] L Wang R Pan X Li and T Jiang ldquoA Bayesian reliability eval-uation method with integrated accelerated degradation testingand field informationrdquo Reliability Engineering amp System Safetyvol 112 pp 38ndash47 2013

[19] Z-S Ye N Chen and Y Shen ldquoA new class of Wiener processmodels for degradation analysisrdquo Reliability Engineering ampSystem Safety vol 139 pp 58ndash67 2015

[20] G A Whitmore and F Schenkelberg ldquoModelling accelerateddegradation data using wiener diffusion with a time scaletransformationrdquo Lifetime Data Analysis vol 3 no 1 pp 27ndash451997

[21] K A Doksum and A Hoyland ldquoModels for variable-stressaccelerated life testing experiments based on Wiener processesand the inverse Gaussian distributionrdquo Technometrics vol 34no 1 pp 74ndash82 1992

[22] H Liao and E A Elsayed ldquoReliability inference for field con-ditions from accelerated degradation testingrdquo Naval ResearchLogistics vol 53 no 6 pp 576ndash587 2006

[23] H-WWang T-X Xu andW-YWang ldquoRemaining life predic-tion based on wiener processes with ADT prior informationrdquoQuality and Reliability Engineering International vol 32 no 3pp 753ndash765 2015

[24] C-Y Peng ldquoInverse Gaussian processes with random effectsand explanatory variables for degradation datardquo Technometricsvol 57 no 1 pp 100ndash111 2015

[25] X Pan XHuang Y Chen YWang and R Kang ldquoConnotationof failure mechanism consistency and identificationmethod foraccelerated testingrdquo in Proceedings of the Prognostics and SystemHealth Management Conference (PHM-Shenzhen rsquo11) May 2011

[26] W B Nelson Accelerated Testing Statistical Models Test Plansand Data Analysis John Wiley amp Sons 2009

[27] Z-S Ye Y Hong and Y Xie ldquoHow do heterogeneities in oper-ating environments affect field failure predictions and testplanningrdquo The Annals of Applied Statistics vol 7 no 4 pp2249ndash2271 2013

Mathematical Problems in Engineering 13

[28] Z-X Zhang X-S Si C-H Hu Q Zhang T Li and C XuldquoPlanning repeated degradation testing for products with three-source variabilityrdquo IEEE Transactions on Reliability vol 65 no2 pp 640ndash647 2016

[29] C-Y Peng and S-T Tseng ldquoMis-specification analysis of lineardegradation modelsrdquo IEEE Transactions on Reliability vol 58no 3 pp 444ndash455 2009

[30] X-S Si W Wang C-H Hu D-H Zhou and M G PechtldquoRemaining useful life estimation based on a nonlinear diffu-sion degradation processrdquo IEEE Transactions on Reliability vol61 no 1 pp 50ndash67 2012

[31] X-S Si W Wang C-H Hu M-Y Chen and D-H Zhou ldquoAWiener-process-based degradationmodel with a recursive filteralgorithm for remaining useful life estimationrdquo MechanicalSystems and Signal Processing vol 35 no 1-2 pp 219ndash237 2013

[32] C-C Tsai S-T Tseng and N Balakrishnan ldquoMis-specificationanalyses of gamma andWiener degradation processesrdquo Journalof Statistical Planning and Inference vol 141 no 12 pp 3725ndash3735 2011

[33] F Haghighi and S J Bae ldquoReliability estimation from lineardegradation and failure time data with competing risks under astep-stress accelerated degradation testrdquo IEEE Transactions onReliability vol 64 no 3 pp 960ndash971 2015

[34] Z-S Ye Y Wang K-L Tsui and M Pecht ldquoDegradation dataanalysis usingwiener processeswithmeasurement errorsrdquo IEEETransactions on Reliability vol 62 no 4 pp 772ndash780 2013

[35] J Huang D S Golubovic S Koh et al ldquoDegradation modelingof mid-power white-light LEDs by using Wiener processrdquoOptics Express vol 23 no 15 pp A966ndashA978 2015

[36] A Beskos O Papaspiliopoulos G O Roberts and P Fearn-head ldquoExact and computationally efficient likelihood-basedestimation for discretely observed diffusion processes (withdiscussion)rdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 68 no 3 pp 333ndash382 2006

[37] J Shang and J E Cavanaugh ldquoAn assumption for the develop-ment of bootstrap variants of the Akaike information criterionin mixed modelsrdquo Statistics and Probability Letters vol 78 no12 pp 1422ndash1429 2008

[38] W QMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[39] W J Padgett andM A Tomlinson ldquoInference from accelerateddegradation and failure data based on Gaussian process mod-elsrdquo Lifetime Data Analysis vol 10 no 2 pp 191ndash206 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Taking the first partial derivatives of the log-likelihoodfunction with respect to 120583120578 1205902120578 yields

120597 ln 119871 (Θ | 119884)120597120583120578= 11205902120578 (

119870sum119896=1

119873119896sum119894=1

1205891198961198791015840119896119894Σminus1119896119894 119910119896119894 minus 120583120578 119870sum119896=1

119873119896sum119894=1

12058921198961198791015840119896119894Σminus1119896119894 119879119896119894) 120597 ln 119871 (Θ | 119884)1205971205902120578 = minus 121205902120578

119870sum119896=1

119873119896sum119894=1

119872119896119894 + 12 (1205902120578)2sdot 119870sum119896=1

119873119896sum119894=1

(119910119896119894 minus 120583120578120589119896119879119896119894)1015840 Σminus1119896119894 (119910119896119894 minus 120583120578120589119896119879119896119894)

(23)

For the special value of ( 120573 2120576) setting the derivation of120597 ln 119871(Θ | 119884)120597120583120578 120597 ln 119871(Θ | 119884)1205971205902120578 to zero the MLE for120583120578 1205902120578 can be expressed as

120578 = sum119870119896=1sum119873119896119894=1 1205891198961198791015840119896119894Σminus1119896119894 119910119896119894sum119870119896=1sum119873119896119894=1 12058921198961198791015840119896119894Σminus1119896119894 119879119896119894 2120578 = 1sum119870119896=1sum119873119896119894=1119872119896119894

sdot 119870sum119896=1

119873119896sum119894=1

(119910119896119894 minus 120578120589119896119879119896119894)1015840 Σminus1119896119894 (119910119896119894 minus 120578120589119896119879119896119894) (24)

Substituting (24) into (22) and simplifying the profilelog-likelihood function can be written as

ln 119871 (Θ | 119884) = minus12 (ln (2120587) + 1) 119870sum119896=1

119873119896sum119894=1

119872119894119896minus 12 ln (2120578) 119870sum

119896=1

119873119896sum119894=1

119872119896119894minus 12 119870sum119896=1

119873119896sum119894=1

10038161003816100381610038161003816Σ11989611989410038161003816100381610038161003816 (25)

The MLE of 120573 and 2120576 can be obtained by maximizingthe profile log-likelihood function in (25) through a three-dimensional search Then substitute them into (24) we canobtain MLE for 120583120578 and 1205902120578

The value of and 2120576 can be obtained by the followingequations

= 120581 sdot 21205782120576 = 1205902120576 sdot 2120578 (26)

42 Parameter Estimation of SSADT The degradation pro-cess of SSADT shown as (18) is relatively complicated Thuswe introduce the accelerated factor first for the sake ofsimplicity The variable 120577119896 is represented by the product of 1205771and the accelerated factor 1205721198961

120589119896 = 12058911205721198961 (27)

The SSADP can be rewritten as

119909119896119894119895

=

12057811989412058911205961119894119895 + radic1205811205891119861 (12059612119895) 1205911198940 le 120596119894119896119895 le 12059111989411205781198941205891 (1205911119894 + 12057221 (1205962119894119895 minus 1205911119894)) + radic1205811205891119861 (1205911119894 + 12057221 (1205962119894119895 minus 1205911119894)) 1205911119894 le 1205962119894119895 le 1205912119894sdot sdot sdot1205781198941205891(119896minus1sum

119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119896119894119895 minus 120591(119896minus1)119894)) + radic1205811205891119861(119896minus1sum119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119896119894119895 minus 120591(119896minus1)119894)) 120591(119896minus1)119894 le 120596119896119894119895 le 120591119896119894sdot sdot sdot1205781198941205891(119870minus1sum

119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119870119894119895 minus 120591(119870minus1)119894)) + radic1205811205891119861(119870minus1sum119899=1

1205721198991 (120591119899119894 minus 120591(119899minus1)119894) + 12057221 (120596119870119894119895 minus 120591(119870minus1)119894)) 120591(119870minus1)119894 le 120596119870119894119895 le 120591119870119894

(28)

Specify 120594119894119896119895 = sum119896minus1119899=1 1205721198991(120591119894119899 minus 120591119894(119899minus1)) + 12057221(120596119894119896119895 minus 120591119894(119896minus1))then the SSADP can be expressed a general formula asfollows

119909119896119894119895 = 1205781198941205891120594119896119894119895 + radic1205811205891119861 (120594119896119894119895) 120591(119896minus1)119894 le 120596119896119894119895 ge 120596119896119894119896 = 1 119870 119894 = 1 119873 119895 = 1 119872119896119894(29)

The foregoing transformation is equivalent to convertingthe degradation driven by stress 119878119896 to the degradation understress 1198781 in physics and the converted equivalent time is 120594119896119894119895At the moment the matrix of degradation and equivalenttransformed time are still bidimensional Because of thecontinuity of the degradation process the matrix can bewritten as a column vector with119872119894 element where119872119894 equalsthe sum of119872119894119896 119896 = 1 119870 and the subscript can be written

Mathematical Problems in Engineering 7

as 119898 for the 119894th product 119898 = 1 119872119894 Then the observeddegradation can be expressed as

119910119894119898 = 119909119894119898 + 120576119894119898119909119894119898 = 1205781198941205891120594119894119898 + 1205811205891119861 (120594119894119898) 1 le 119898 le 119870sum

119896=1

119872119894119896(30)

Define Δ1205941198941 = 1205941198941 Δ120594119894119898 = 120594119894119898 minus 120594119894(119898minus1) Δ120594119894 =(Δ1205941198941 Δ120594119894119872119894)1015840 119877119894 = Δ120594119894 and Δ119910119894119898 = 119910119894119898 minus 119910119894(119898minus1) for119894 = 1 119873 119898 = 1 119872119894 Then Δ119910119894 = (Δ1199101198941 Δ119910119894119872119894)1015840follows a multivariate normal distribution 119873(1205831205781205891119877119894 Π119894)where

Π119894 = 119882119894 + 1205902120578120589211198771198941198771015840119894 (119882119901119902)119894 = cov (Δ119910119894119901 Δ119910119894119902)

=

1205811205891119877119894119901 + 1205902120576 119901 = 119902 = 11205811205891119877119894119901 + 21205902120576 119901 = 119902 gt 1minus1205902120576 119901 = 119902 + 1 or 119901 = 119902 minus 10 otherwise

(31)

So the log-likelihood function can be expressed as

ln 119871 (Θ | 119884)= minus12 ln (2120587) 119873sum

119894=1

119872119894 minus 12 119873sum119894=1

ln (1003816100381610038161003816Π1198941003816100381610038161003816)minus 12 119873sum119894=1

(Δ119910119894 minus 1205831205781205891119877119894)1015840Πminus1119894 (Δ119910119894 minus 1205831205781205891119877119894) (32)

where1003816100381610038161003816Π1198941003816100381610038161003816 = 10038161003816100381610038161198821198941003816100381610038161003816 (1 + 1205902120578120589211198771015840119894119882minus1119894 119877119894)Πminus1119894 = 119882minus1119894 minus 1205902120578120589211 + 1205902120578120589211198771015840119894119882minus1119894 119877119894119882minus1119894 1198771198941198771015840119894119882minus1119894 (33)

Taking the first partial derivatives of the log-likelihoodfunction with respect to 120583120578 1205902120578 yields

120597 ln 119871 (Θ | 119884)120597120583120578 = 1205891 119873sum119894=1

1198771015840119894Πminus1119894 Δ119910119894 minus 12058312057812058921 119873sum119894=1

1198771015840119894Πminus1119894 119877119894 (34)

120597 ln 119871 (Θ | 119884)1205971205902120578 = minus 119873sum119894=1

120589211198771015840119894119882minus1119894 1198771 + 1205902120578120589211198771015840119894119882minus1119894 119877119894+ 119873sum119894=1

12058921 (Δ119910119894 minus 1205831205781205891119877119894)1015840119882minus1119894 1198771198941198771015840119894119882minus1119894 (Δ119910119894 minus 1205831205781205891119877119894)(1 + 1205902120578120589211198771015840119894119882minus1119894 119877)2(35)

For the special value of (120581 120573 1205902120576 ) setting the derivation of120597 ln 119871(Θ | 119884)120597120583120578 to zero the restricted MLE for 120583120578 can beexpressed as

120578 = sum119873119894=1 1198771015840119894Πminus1119894 Δ1199101198941205891sum119873119894=1 1198771015840119894Πminus1119894 119877119894 (36)

Similarly set the derivation of 120597 ln 119871(Θ | 119884)120597120583120578 to zeroand the results of the MLE for 1205902120578 will be discussed on twocases

Case 1 It was assumed that the number of the measurementsand the measurement points of each sample are the same forall of the samples under all of the accelerated stressThat is tosay the subscript of 119877119894 Π119894 and119882119894 can be removed

Thus the restricted MLE for 120583120578 can be expressed as

120578 = sum119873119894=1 1198771015840Πminus1119894 Δ11991011989412058911198731198771015840Πminus1119894 119877 (37)

The first partial derivatives of the log-likelihood functionto 1205902120578 can be rewritten as

120597 ln 119871 (Θ | 119884)1205971205902120578 = minus119873 120589211198771015840119882minus11198771 + 1205902120578120589211198771015840119882minus1119877+ 12058921(1 + 1205902120578120589211198771015840119882minus1119877)2sdot 119873sum119894=1

(Δ119910119894 minus 1205831205781205891119877)119882minus11198771198771015840119882minus1 (Δ119910119894 minus 1205831205781205891119877)1015840 (38)

For the special value of (120581 120573 1205902120576 ) setting the derivation of120597 ln 119871(Θ | 119884)1205971205902120578 to zero the result of the MLE for 1205902120578 can beexpressed as2120578

= sum119873119894=1 (Δ119910119894 minus 1205831205781205891119877)119882minus11198771198771015840119882minus1 (Δ119910119894 minus 1205831205781205891119877)101584011987312058921 (1198771015840119882minus1119877)2minus 1120589211198771015840119882minus1119877

(39)

Substituting (37) (39) into (32) and simplifying the profilelog-likelihood function can be written as

ln 119871 (120581 120573 1205902120576 | 120578 2120578 119884) = minus1198721198732 ln (2120587) minus 1198732 minus 1198732sdot ln (|119882|)minus 12

119873sum119894=1

Δ1199101015840119894119882minus1Δ119910119894 minus sum119873119894=1 (1198771015840119882minus1Δ119910119894)21198771015840119882minus1119877 minus 12sdot ln

sum119873119894=1 (1198771015840119882minus1Δ119910119894)21198731198771015840119882minus1119877 minus sum119873119894=1 (1198771015840119882minus1Δ119910119894)211987321198771015840119882minus1119877

(40)

8 Mathematical Problems in Engineering

where119872 is themeasurement time point of each samplewhichis the same for all of the samples based on the assumption ofCase 1

The MLE of 120581 120573 and 1205902120576 can be obtained by maximizingthe profile log-likelihood function in (40) through a three-dimensional searchThen substitute them into (37) and (39)we can obtain the MLE for 120583120578 and 1205902120578 Case 2 The number of the measurements and the measure-ment points of each sample are different for all of the samplesunder all of the accelerated stress In this case the first partialderivatives of the log-likelihood function to 1205902120578 are shown asin (35)There may be no analytical form by setting the partialderivatives to zero The evaluation of 120583120578 can be expressed as(36) Then the profile log-likelihood function of 1205902120578 120581 120573 1205902120576can be written as

ln 119871 (120581 120573 120590120576 1205902120578 | 120578 119884)= minus12 ln (2120587) 119873sum

119894=1

119872119894 minus 12 119873sum119894=1

ln (1003816100381610038161003816Π1198941003816100381610038161003816)minus 119873sum119894=1

(Δ119910119894 minus 1205781205891119877119894)Πminus1119894 (Δ119910119894 minus 1205781205891119877119894)1015840 (41)

The MLE of 1205902120578 120581 120573 and 1205902120576 can be obtained by maxi-mizing the profile log-likelihood function in (41) through afour-dimensional search Then substitute them into (36) wecan obtain the MLE for 120583120578

It is not to say that we can only use degradation forCSADT and increment for SSADT but just make an intro-duction to both of the two methods in the limited space Inaddition we could verify the results by comparing the esti-mation calculated by the two methods to avoid computationerrors

5 Simulation Data Analysis

In order to validate the model described before and theparameter estimation methods simulation test was con-ducted

51 The Simulation Method of CSADT Data The parametersΘ = 120583120578 1205902120578 120581 120573 1205902120576 should be given before the simulationand simulation process is shown as in the following steps

Step 1 Set 119896 = 1Step 2 Generate 119873119894 random numbers with subscripts 119894 119894 =1 2 119873119896 subject to normal distribution with mean 120583120578 andvariance 1205902120578 Step 3 Let 119897 = 0Step 4 Calculate119883119896119894(119897+1) using the Euler approximation [36]

119883119896119894(119897+1) = 119883119896119894119897 + 120578119894120589119896120596 (Δ119905) + 120581120589119896radic120596 (Δ119905)Ψ (42)

where Ψ is the random number according to the standardnormal distribution and Δ119905 is the step size in simulation and1198831198961198940 = 0 As such set 119897 = 119897 + 1 and return to Step 4 to get thedegradation path until 119897 = 119871119896119894+1 where 119871119896119894 is the simulationsteps for 119894th under stress 119878119896Step 5 Set 119896 = 119896 + 1 and return to Step 2 until 119896 = 119870 + 1Step 6 Extract the degradation based on the predefinedmeasurement time point 119905119896119894119895 where 119896 = 1 2 119870 119894 =1 2 119873119896 and 119895 = 1 2 119872119896119894 from the data set 119883119896119894119897 andget the data set of degradation119883119896119894119895Step 7 Simulate the measured degradation 119884119896119894119895 = 119883119896119894119895 + 120576119896119894119895where 120576119896119894119895 sim 119873(0 1205902120576 )52 The Simulation Method of SSADT Data There are somedifferences in the simulation process for SSADT comparedwith CSADT The simulation process is shown as follows

Step 1 Generate 119873 random numbers with subscripts 119894 119894 =1 2 119873 subject to normal distribution with mean 120583120578 andvariance 1205902120578 Step 2 Let 119896 = 1Step 3 Set 119897 = 1Step 4 Calculate119883119896119894(119897+1)

119883119896119894(119897+1) = 119883119896119894119897 + 120578119894120589119896120596 (Δ119905) + 120581120589119896radic120596 (Δ119905)Ψ (43)

where Ψ is the random number according to the standardnormal distribution and Δ119905 is the step size in simulation and1198831198961198940 = 01198831198961198940 = 119883(119896minus1)119894119871(119896minus1) (119896 gt 1) As such set 119897 = 119897+1 andreturn to Step 4 to get the degradation path until 119897 = 119871119896119894 + 1Step 5 Set 119896 = 119896 + 1 and return to Step 2 until 119896 = 119870 + 1Step 6 Extract the degradation based on the predefinedmeasurement time point 119905119896119894119895 where 119896 = 1 2 119870 119894 =1 2 119873 and 119895 = 1 2 119872119894119896 from the data set 119883119896119894119897 andget the data set of degradation119883119896119894119895Step 7 Simulate the measured degradation 119884119896119894119895 = 119883119896119894119895 + 120576119896119894119895where 120576119896119894119895 sim 119873(0 1205902120576 )53 The Analysis of the Simulated SSADT Data We justgive the analysis of simulated SSADT data here becausewe would give a case study of the CSADT later It wasassumed that the accelerated stress is temperature and thesimulation test contains 4 stresseswhich are 50∘C 60∘C 70∘Cand 80∘C For simplicity the transformed time function isset as Λ(119905) = 119905 and 20 degradation paths are generatedThen the parameters for degradation process are assumedas Θ = 5 4 025 minus3000 001 Moreover the time intervalmeasurement is 25 h for each sample of the whole simulationtest with 40 measurement time points The degradation dataare depicted as in Figure 1

Mathematical Problems in Engineering 9

Table 2 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 log-LF AIC RETruth value 5 1 025 mdash minus3000 001 mdash mdash mdash1198720 501 139 028 mdash minus297561 00078 142228 minus283456 084021198721 512 175 mdash 12907 lowast 10minus4 minus298081 00129 112289 minus223577 562711198722 620 mdash mdash 15213 lowast 10minus4 minus305175 13789 lowast 10minus9 109361 minus217721 78049

0 200 400 600 800 1000

0

05

1

15

2

25

th

y

Figure 1 The simulation degradation paths of SSADT

For simplicity the degradation model for SSADT pro-posed in this paper is referred to as1198720 the model presentedby Tang et al [17] as 1198721 and the stochastic SSADT modeldescribed in reference [13] as 1198722 The estimation results ofthe unknown parameters the log-likelihood function valueand the Akaike information criterion (AIC) [37] which isevaluated by (44) are summarized in Table 2 As we cansee from Table 2 our model clearly outperforms model interms of the log-LF and AIC The estimates of 120583120578 1205902120578 and1205902120576 are more close to the truth values compared with 1198721and 1198722 Meanwhile the diffusion-related variables includeparameters 120581 and 120573 and the diffusion parameter under thefour stresses can be obtained by (10) which is (28377 lowast 10minus548534 lowast 10minus5 78244 lowast 10minus5 12001 lowast 10minus4) in 1198721 and(22334lowast10minus5 39912lowast10minus5 64597lowast10minus5 99423lowast10minus5) forthe true model Moreover taking 1198781 as a reference a criterionformulizing the relative error (RE) is shown as (45) It is clearthat1198720 has the smallest RE from Table 2

AIC (119873) = minus2 ln 119871 (Θ) + 2119873 (44)

RE = 1003816100381610038161003816100381610038161003816100381610038161003816120583120578 minus 120578120578

1003816100381610038161003816100381610038161003816100381610038161003816 +100381610038161003816100381610038161003816100381610038161003816100381610038161205902120578 minus 21205782120578

10038161003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161003816(1205902119861)1 minus (2119861)1(2119861)1

10038161003816100381610038161003816100381610038161003816100381610038161003816+ 1003816100381610038161003816100381610038161003816100381610038161003816120573 minus

1003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161205902120576 minus 21205762120576

1003816100381610038161003816100381610038161003816100381610038161003816 (45)

It is assumed that the failure threshold119908 is equal to 5 andthe normal stress is 25∘C The PDF and CDF under normalstress are shown as in Figure 2 The phenomenon can beobserved where the PDF and CDF of our model most nearlyapproached the real model By contrasting 1198721 with 1198722 itcan be concluded that the neglect of random effects wouldresult in sharper PDF which is consistent with the result inreference [17] And it is noteworthy that the 05 quantiles ofthe failure life are almost equal from the CDF of the threemodels But there is a greater difference between the realmodel and 1198721 while the indicator is taken as mean time tofailure as well as the difference between the real model and1198722 So the correlation between the stress and the diffusionparameter should not be neglected Next we are going toprove the superiority of our model in practical application

6 Case Application

The CSADT model with covariates and random effects isverified by the accelerated degradation data of carbon-filmresistors whose raw data set is explicitly given in Table C3of Meeker and Escobar [38]There were 29 samples and threeaccelerated temperature stresses in the whole test where ninesamples were observed at 83∘C the remaining two stresses133∘C and 173∘C had 10 samples for each stress All of thesamples were observed at the same time points with 1199050 = 01199051 = 452 1199052 = 1030 1199053 = 4341 and 1199054 = 8084 (inhours) Similar to [39] it was assumed that the standardoperating temperature was 50∘C and the threshold value forpercent increase in resistance was taken to be 119862 = 12 Itis reasonable and computationally easier to use the ratios inresistance to the initial value for each rather than the percentincrease [9] For ease of calculation the logarithm of the ratiowas taken as the degradation here That is the value 028of the percent increase was changed to 2796119864 minus 3 and soon and the transformed threshold value 119908 was 01133 Thedegradation paths of all the sample were as shown in Figure 3It can be seen that the degradation of the samples uniformlyshowed a nonlinear characteristic especially at the beginningof the ADT Thus the degradation is modeled with the timescale transformedWiener process as (1) with 120596 = Λ(119905) = 119905119887

Similarly the proposed degradation model for SSADT inthis paper is referred to as1198720 themodel presented by Tang etal [17] as1198721 and themodel described in reference [9] as1198723The estimation results of the unknown parameters the log-likelihood function value and the AIC are shown in Table 3The estimated parameters 119887 are approximately equal for thethree models Take 119887 = 050 as an example to show the time

10 Mathematical Problems in Engineering

Table 3 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 b Log-LF AIC MTTF1198720 823 207 108119864 minus 02 mdash minus420282 182119864 minus 20 050 51848 minus102497 380119864 + 071198721 1471 433 mdash 413119864 minus 07 minus458656 506119864 minus 20 053 48637 minus96073 515119864 + 071198722 1111 mdash mdash 676119864 minus 07 minus447982 411119864 minus 18 053 46837 minus92674 406119864 + 07

0 1 2 3 4 50

05

1

15

2

25

th

The P

DF

Real model

times104

times10minus4

M1

M0

M2

0 1 2 3 4 50

02

04

06

08

1

th

The C

DF

times104

Real modelM1

M0

M2

Figure 2 The comparison of the PDF and CDF of the three models for simulated SSADT data

0 2000 4000 6000 80000

0002

0004

0006

0008

001

th

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 2000 4000 6000 80000

001

002

003

th

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 2000 4000 6000 8000th

0

002

004

006

008

01

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 3 The degradation paths of carbon-film resistors

scale transformed degradation paths as in Figure 4 which areapproximately straight lines compared with Figure 3

Our model has the largest log-LF and smallest AICcompared with1198721 and1198722 The result proves that our modelis more suitable for the degradation data Besides 1198721 hasthe larger log-LF and smaller AIC compared with1198722 So therandom effects could not be neglected It can be observed thatthe variance of the drift parameter is slightly largerwhichmaybe owing to the fixed value of diffusion parameter in1198721

The PDF and CDF under the standard operating temper-ature are as shown in Figure 5The time corresponding to thepeak values of PDF were as follows 2676 lowast 107 1049 lowast 107

and 5028lowast106 for1198720 to1198722 But theMTTF of1198720 isminimalfrom Table 3 And the uncertainty in the estimated PDFs ofthe lifetimes under1198720 is smallest compared to1198721 and1198722 asseen in Figure 5When time 119905 approaches positive infinity theCDF of ourmodel which is calculated by (13) tends illimitablyto 1 and the CDF of 1198721 is roughly equal to 09997 It followsthat our model could do well in distinguishing the differencefrom random effects and the effect of covariates Anotherthing which is worth noting is that the measurement error isquite tiny in the case but it is just an individual phenomenonThemeasurement error should also be considered in practicalapplication

Mathematical Problems in Engineering 11

0 20 40 60 800

0002

0004

0006

0008

001

Λ(t)

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 20 40 60 800

001

002

003

Λ(t)

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 20 40 60 800

002

004

006

008

01

Λ(t)

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 4 The degradation paths of carbon-film resistors under transformed time scale

0 2 4 6 8 100

05

1

15

2

25

3

35

th

times10minus8

M0

M1

M2

The P

DF

times107

0 2 4 6 8 100

02

04

06

08

1

th

M0

M1

M2

The C

DF

times107

Figure 5 The comparison of the PDF and CDF of the three models for real CSADT data

7 Conclusion

In this paper the degradation models based on nonlinearWiener process are established for both constant stress accel-erated degradation data and step stress accelerated degrada-tion data Before the establishment the relationship betweenthe drift parameter and stress variables is derived based onthe invariance principle of failure mechanism and Nelsonassumption so is the relationship between the diffusionparameter and stress variables It is concluded that the ratioof drift parameters under two stresses is a constant which isirrelevant to the testing time and depends only on the twostresses as long as the ratio of diffusion parameters is equalto the ratio of drift parameters And the ratio is defined asaccelerated factor Besides the random effects are also takeninto consideration where the drift parameter is assumed to be

normally distributed and the diffusion parameter is same forall of the samples under a certain stress Then the PDF andCDF of the FHT are deduced considering random effects

Because of the dependency between the diffusion param-eter and stress variables the degradation process is quitedifferent either for CSADT or for SSADT The CSADP andSSADP with random effects are modeled Moreover theunknown parameters are solved by MLE based on the twoproperties of Wiener process At the end of the paper thesimulated data of SSADT and the CSADTdata of carbon-filmresistors are both analyzed to verify the proposed model Itis concluded that the model has the biggest log-LF and thesmallest AIC compared with the two other models

The innovation of this paper lies in the following Firstthe random effects are considered under the new relationshipbetween the diffusion parameter and accelerated stresses

12 Mathematical Problems in Engineering

Second the degradation process was modeled for bothCSADT and SSADT Thirdly the unknown parameters wereestimated based on the two properties of Wiener process andthe result of theMLE for 1205902120578 is discussed on two cases Fourththe measurement error of the degradation data is also valued

However we have only considered the random effects ofthe drift parameter in this paper due to the complexity of thecomputation A further research may consider the randomeffects of the diffusion parameter into the model At the sametime the study of the paper may provide new ideas for therelativity analysis between the parameters of other stochasticprocess and stress variables

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by Jiangsu Province GraduateStudent Scientific Research Innovation Project of China(Project KYLX15 0330)The help is gratefully acknowledged

References

[1] H-F Yu ldquoOptimal classification of highly-reliable productswhose degradation paths satisfyWiener processesrdquo EngineeringOptimization vol 35 no 3 pp 313ndash324 2003

[2] XWang N Balakrishnan and B Guo ldquoResidual life estimationbased on a generalized Wiener degradation processrdquo ReliabilityEngineering and System Safety vol 124 pp 13ndash23 2014

[3] Y Ren Q Feng T Ye and B Sun ldquoA novel model of reliabilityassessment for circular electrical connectorsrdquo IEEE Transac-tions on Components Packaging andManufacturing Technologyvol 5 no 6 pp 755ndash761 2015

[4] Z-S Ye and M Xie ldquoStochastic modelling and analysis ofdegradation for highly reliable productsrdquo Applied StochasticModels in Business and Industry vol 31 no 1 pp 16ndash32 2015

[5] H Lim and B-J Yum ldquoOptimal design of accelerated degrada-tion tests based on Wiener process modelsrdquo Journal of AppliedStatistics vol 38 no 2 pp 309ndash325 2011

[6] C J Lu and W Q Meeker ldquoUsing degradation measures toestimate a time-to-failure distributionrdquo Technometrics vol 35no 2 pp 161ndash174 1993

[7] G AWhitmore ldquoEstimating degradation by a wiener diffusionprocess subject to measurement errorrdquo Lifetime Data Analysisvol 1 no 3 pp 307ndash319 1995

[8] Z Pan and N Balakrishnan ldquoReliability modeling of degra-dation of products with multiple performance characteristicsbased on gamma processesrdquo Reliability Engineering amp SystemSafety vol 96 no 8 pp 949ndash957 2011

[9] C Park and W J Padgett ldquoAccelerated degradation modelsfor failure based on geometric Brownian motion and gammaprocessesrdquo Lifetime Data Analysis vol 11 no 4 pp 511ndash5272005

[10] Z-S Ye and N Chen ldquoThe inverse Gaussian process as adegradation modelrdquo Technometrics vol 56 no 3 pp 302ndash3112014

[11] A Onar and W J Padgett ldquoAccelerated test models with theinverse Gaussian distributionrdquo Journal of Statistical Planningand Inference vol 89 no 1-2 pp 119ndash133 2000

[12] T-R Tsai Y L Lio and N Jiang ldquoOptimal decisions on theaccelerated degradation test plan under the Wiener processrdquoQuality Technology and Quantitative Management vol 11 no4 pp 461ndash470 2014

[13] C-M Liao and S-T Tseng ldquoOptimal design for step-stressaccelerated degradation testsrdquo IEEE Transactions on Reliabilityvol 55 no 1 pp 59ndash66 2006

[14] L C Tang G Yang and M Xie ldquoPlanning of step-stress accel-erated degradation testrdquo in Proceedings of the Annual Reliabil-ity and Maintainability SymposiummdashProceedings InternationalSymposium on Product Quality and Integrity pp 287ndash292January 2004

[15] C-H Hu M-Y Lee and J Tang ldquoOptimum step-stress accel-erated degradation test for Wiener degradation process underconstraintsrdquo European Journal of Operational Research vol 241no 2 pp 412ndash421 2015

[16] S Tang ldquoStep stress accelerated degradation process modelingand remaining useful life estimationrdquo Journal of MechanicalEngineering vol 50 no 16 p 33 2014

[17] S Tang X Guo C Yu H Xue and Z Zhou ldquoAccelerated degra-dation tests modeling based on the nonlinear wiener processwith random effectsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 560726 11 pages 2014

[18] L Wang R Pan X Li and T Jiang ldquoA Bayesian reliability eval-uation method with integrated accelerated degradation testingand field informationrdquo Reliability Engineering amp System Safetyvol 112 pp 38ndash47 2013

[19] Z-S Ye N Chen and Y Shen ldquoA new class of Wiener processmodels for degradation analysisrdquo Reliability Engineering ampSystem Safety vol 139 pp 58ndash67 2015

[20] G A Whitmore and F Schenkelberg ldquoModelling accelerateddegradation data using wiener diffusion with a time scaletransformationrdquo Lifetime Data Analysis vol 3 no 1 pp 27ndash451997

[21] K A Doksum and A Hoyland ldquoModels for variable-stressaccelerated life testing experiments based on Wiener processesand the inverse Gaussian distributionrdquo Technometrics vol 34no 1 pp 74ndash82 1992

[22] H Liao and E A Elsayed ldquoReliability inference for field con-ditions from accelerated degradation testingrdquo Naval ResearchLogistics vol 53 no 6 pp 576ndash587 2006

[23] H-WWang T-X Xu andW-YWang ldquoRemaining life predic-tion based on wiener processes with ADT prior informationrdquoQuality and Reliability Engineering International vol 32 no 3pp 753ndash765 2015

[24] C-Y Peng ldquoInverse Gaussian processes with random effectsand explanatory variables for degradation datardquo Technometricsvol 57 no 1 pp 100ndash111 2015

[25] X Pan XHuang Y Chen YWang and R Kang ldquoConnotationof failure mechanism consistency and identificationmethod foraccelerated testingrdquo in Proceedings of the Prognostics and SystemHealth Management Conference (PHM-Shenzhen rsquo11) May 2011

[26] W B Nelson Accelerated Testing Statistical Models Test Plansand Data Analysis John Wiley amp Sons 2009

[27] Z-S Ye Y Hong and Y Xie ldquoHow do heterogeneities in oper-ating environments affect field failure predictions and testplanningrdquo The Annals of Applied Statistics vol 7 no 4 pp2249ndash2271 2013

Mathematical Problems in Engineering 13

[28] Z-X Zhang X-S Si C-H Hu Q Zhang T Li and C XuldquoPlanning repeated degradation testing for products with three-source variabilityrdquo IEEE Transactions on Reliability vol 65 no2 pp 640ndash647 2016

[29] C-Y Peng and S-T Tseng ldquoMis-specification analysis of lineardegradation modelsrdquo IEEE Transactions on Reliability vol 58no 3 pp 444ndash455 2009

[30] X-S Si W Wang C-H Hu D-H Zhou and M G PechtldquoRemaining useful life estimation based on a nonlinear diffu-sion degradation processrdquo IEEE Transactions on Reliability vol61 no 1 pp 50ndash67 2012

[31] X-S Si W Wang C-H Hu M-Y Chen and D-H Zhou ldquoAWiener-process-based degradationmodel with a recursive filteralgorithm for remaining useful life estimationrdquo MechanicalSystems and Signal Processing vol 35 no 1-2 pp 219ndash237 2013

[32] C-C Tsai S-T Tseng and N Balakrishnan ldquoMis-specificationanalyses of gamma andWiener degradation processesrdquo Journalof Statistical Planning and Inference vol 141 no 12 pp 3725ndash3735 2011

[33] F Haghighi and S J Bae ldquoReliability estimation from lineardegradation and failure time data with competing risks under astep-stress accelerated degradation testrdquo IEEE Transactions onReliability vol 64 no 3 pp 960ndash971 2015

[34] Z-S Ye Y Wang K-L Tsui and M Pecht ldquoDegradation dataanalysis usingwiener processeswithmeasurement errorsrdquo IEEETransactions on Reliability vol 62 no 4 pp 772ndash780 2013

[35] J Huang D S Golubovic S Koh et al ldquoDegradation modelingof mid-power white-light LEDs by using Wiener processrdquoOptics Express vol 23 no 15 pp A966ndashA978 2015

[36] A Beskos O Papaspiliopoulos G O Roberts and P Fearn-head ldquoExact and computationally efficient likelihood-basedestimation for discretely observed diffusion processes (withdiscussion)rdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 68 no 3 pp 333ndash382 2006

[37] J Shang and J E Cavanaugh ldquoAn assumption for the develop-ment of bootstrap variants of the Akaike information criterionin mixed modelsrdquo Statistics and Probability Letters vol 78 no12 pp 1422ndash1429 2008

[38] W QMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[39] W J Padgett andM A Tomlinson ldquoInference from accelerateddegradation and failure data based on Gaussian process mod-elsrdquo Lifetime Data Analysis vol 10 no 2 pp 191ndash206 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

as 119898 for the 119894th product 119898 = 1 119872119894 Then the observeddegradation can be expressed as

119910119894119898 = 119909119894119898 + 120576119894119898119909119894119898 = 1205781198941205891120594119894119898 + 1205811205891119861 (120594119894119898) 1 le 119898 le 119870sum

119896=1

119872119894119896(30)

Define Δ1205941198941 = 1205941198941 Δ120594119894119898 = 120594119894119898 minus 120594119894(119898minus1) Δ120594119894 =(Δ1205941198941 Δ120594119894119872119894)1015840 119877119894 = Δ120594119894 and Δ119910119894119898 = 119910119894119898 minus 119910119894(119898minus1) for119894 = 1 119873 119898 = 1 119872119894 Then Δ119910119894 = (Δ1199101198941 Δ119910119894119872119894)1015840follows a multivariate normal distribution 119873(1205831205781205891119877119894 Π119894)where

Π119894 = 119882119894 + 1205902120578120589211198771198941198771015840119894 (119882119901119902)119894 = cov (Δ119910119894119901 Δ119910119894119902)

=

1205811205891119877119894119901 + 1205902120576 119901 = 119902 = 11205811205891119877119894119901 + 21205902120576 119901 = 119902 gt 1minus1205902120576 119901 = 119902 + 1 or 119901 = 119902 minus 10 otherwise

(31)

So the log-likelihood function can be expressed as

ln 119871 (Θ | 119884)= minus12 ln (2120587) 119873sum

119894=1

119872119894 minus 12 119873sum119894=1

ln (1003816100381610038161003816Π1198941003816100381610038161003816)minus 12 119873sum119894=1

(Δ119910119894 minus 1205831205781205891119877119894)1015840Πminus1119894 (Δ119910119894 minus 1205831205781205891119877119894) (32)

where1003816100381610038161003816Π1198941003816100381610038161003816 = 10038161003816100381610038161198821198941003816100381610038161003816 (1 + 1205902120578120589211198771015840119894119882minus1119894 119877119894)Πminus1119894 = 119882minus1119894 minus 1205902120578120589211 + 1205902120578120589211198771015840119894119882minus1119894 119877119894119882minus1119894 1198771198941198771015840119894119882minus1119894 (33)

Taking the first partial derivatives of the log-likelihoodfunction with respect to 120583120578 1205902120578 yields

120597 ln 119871 (Θ | 119884)120597120583120578 = 1205891 119873sum119894=1

1198771015840119894Πminus1119894 Δ119910119894 minus 12058312057812058921 119873sum119894=1

1198771015840119894Πminus1119894 119877119894 (34)

120597 ln 119871 (Θ | 119884)1205971205902120578 = minus 119873sum119894=1

120589211198771015840119894119882minus1119894 1198771 + 1205902120578120589211198771015840119894119882minus1119894 119877119894+ 119873sum119894=1

12058921 (Δ119910119894 minus 1205831205781205891119877119894)1015840119882minus1119894 1198771198941198771015840119894119882minus1119894 (Δ119910119894 minus 1205831205781205891119877119894)(1 + 1205902120578120589211198771015840119894119882minus1119894 119877)2(35)

For the special value of (120581 120573 1205902120576 ) setting the derivation of120597 ln 119871(Θ | 119884)120597120583120578 to zero the restricted MLE for 120583120578 can beexpressed as

120578 = sum119873119894=1 1198771015840119894Πminus1119894 Δ1199101198941205891sum119873119894=1 1198771015840119894Πminus1119894 119877119894 (36)

Similarly set the derivation of 120597 ln 119871(Θ | 119884)120597120583120578 to zeroand the results of the MLE for 1205902120578 will be discussed on twocases

Case 1 It was assumed that the number of the measurementsand the measurement points of each sample are the same forall of the samples under all of the accelerated stressThat is tosay the subscript of 119877119894 Π119894 and119882119894 can be removed

Thus the restricted MLE for 120583120578 can be expressed as

120578 = sum119873119894=1 1198771015840Πminus1119894 Δ11991011989412058911198731198771015840Πminus1119894 119877 (37)

The first partial derivatives of the log-likelihood functionto 1205902120578 can be rewritten as

120597 ln 119871 (Θ | 119884)1205971205902120578 = minus119873 120589211198771015840119882minus11198771 + 1205902120578120589211198771015840119882minus1119877+ 12058921(1 + 1205902120578120589211198771015840119882minus1119877)2sdot 119873sum119894=1

(Δ119910119894 minus 1205831205781205891119877)119882minus11198771198771015840119882minus1 (Δ119910119894 minus 1205831205781205891119877)1015840 (38)

For the special value of (120581 120573 1205902120576 ) setting the derivation of120597 ln 119871(Θ | 119884)1205971205902120578 to zero the result of the MLE for 1205902120578 can beexpressed as2120578

= sum119873119894=1 (Δ119910119894 minus 1205831205781205891119877)119882minus11198771198771015840119882minus1 (Δ119910119894 minus 1205831205781205891119877)101584011987312058921 (1198771015840119882minus1119877)2minus 1120589211198771015840119882minus1119877

(39)

Substituting (37) (39) into (32) and simplifying the profilelog-likelihood function can be written as

ln 119871 (120581 120573 1205902120576 | 120578 2120578 119884) = minus1198721198732 ln (2120587) minus 1198732 minus 1198732sdot ln (|119882|)minus 12

119873sum119894=1

Δ1199101015840119894119882minus1Δ119910119894 minus sum119873119894=1 (1198771015840119882minus1Δ119910119894)21198771015840119882minus1119877 minus 12sdot ln

sum119873119894=1 (1198771015840119882minus1Δ119910119894)21198731198771015840119882minus1119877 minus sum119873119894=1 (1198771015840119882minus1Δ119910119894)211987321198771015840119882minus1119877

(40)

8 Mathematical Problems in Engineering

where119872 is themeasurement time point of each samplewhichis the same for all of the samples based on the assumption ofCase 1

The MLE of 120581 120573 and 1205902120576 can be obtained by maximizingthe profile log-likelihood function in (40) through a three-dimensional searchThen substitute them into (37) and (39)we can obtain the MLE for 120583120578 and 1205902120578 Case 2 The number of the measurements and the measure-ment points of each sample are different for all of the samplesunder all of the accelerated stress In this case the first partialderivatives of the log-likelihood function to 1205902120578 are shown asin (35)There may be no analytical form by setting the partialderivatives to zero The evaluation of 120583120578 can be expressed as(36) Then the profile log-likelihood function of 1205902120578 120581 120573 1205902120576can be written as

ln 119871 (120581 120573 120590120576 1205902120578 | 120578 119884)= minus12 ln (2120587) 119873sum

119894=1

119872119894 minus 12 119873sum119894=1

ln (1003816100381610038161003816Π1198941003816100381610038161003816)minus 119873sum119894=1

(Δ119910119894 minus 1205781205891119877119894)Πminus1119894 (Δ119910119894 minus 1205781205891119877119894)1015840 (41)

The MLE of 1205902120578 120581 120573 and 1205902120576 can be obtained by maxi-mizing the profile log-likelihood function in (41) through afour-dimensional search Then substitute them into (36) wecan obtain the MLE for 120583120578

It is not to say that we can only use degradation forCSADT and increment for SSADT but just make an intro-duction to both of the two methods in the limited space Inaddition we could verify the results by comparing the esti-mation calculated by the two methods to avoid computationerrors

5 Simulation Data Analysis

In order to validate the model described before and theparameter estimation methods simulation test was con-ducted

51 The Simulation Method of CSADT Data The parametersΘ = 120583120578 1205902120578 120581 120573 1205902120576 should be given before the simulationand simulation process is shown as in the following steps

Step 1 Set 119896 = 1Step 2 Generate 119873119894 random numbers with subscripts 119894 119894 =1 2 119873119896 subject to normal distribution with mean 120583120578 andvariance 1205902120578 Step 3 Let 119897 = 0Step 4 Calculate119883119896119894(119897+1) using the Euler approximation [36]

119883119896119894(119897+1) = 119883119896119894119897 + 120578119894120589119896120596 (Δ119905) + 120581120589119896radic120596 (Δ119905)Ψ (42)

where Ψ is the random number according to the standardnormal distribution and Δ119905 is the step size in simulation and1198831198961198940 = 0 As such set 119897 = 119897 + 1 and return to Step 4 to get thedegradation path until 119897 = 119871119896119894+1 where 119871119896119894 is the simulationsteps for 119894th under stress 119878119896Step 5 Set 119896 = 119896 + 1 and return to Step 2 until 119896 = 119870 + 1Step 6 Extract the degradation based on the predefinedmeasurement time point 119905119896119894119895 where 119896 = 1 2 119870 119894 =1 2 119873119896 and 119895 = 1 2 119872119896119894 from the data set 119883119896119894119897 andget the data set of degradation119883119896119894119895Step 7 Simulate the measured degradation 119884119896119894119895 = 119883119896119894119895 + 120576119896119894119895where 120576119896119894119895 sim 119873(0 1205902120576 )52 The Simulation Method of SSADT Data There are somedifferences in the simulation process for SSADT comparedwith CSADT The simulation process is shown as follows

Step 1 Generate 119873 random numbers with subscripts 119894 119894 =1 2 119873 subject to normal distribution with mean 120583120578 andvariance 1205902120578 Step 2 Let 119896 = 1Step 3 Set 119897 = 1Step 4 Calculate119883119896119894(119897+1)

119883119896119894(119897+1) = 119883119896119894119897 + 120578119894120589119896120596 (Δ119905) + 120581120589119896radic120596 (Δ119905)Ψ (43)

where Ψ is the random number according to the standardnormal distribution and Δ119905 is the step size in simulation and1198831198961198940 = 01198831198961198940 = 119883(119896minus1)119894119871(119896minus1) (119896 gt 1) As such set 119897 = 119897+1 andreturn to Step 4 to get the degradation path until 119897 = 119871119896119894 + 1Step 5 Set 119896 = 119896 + 1 and return to Step 2 until 119896 = 119870 + 1Step 6 Extract the degradation based on the predefinedmeasurement time point 119905119896119894119895 where 119896 = 1 2 119870 119894 =1 2 119873 and 119895 = 1 2 119872119894119896 from the data set 119883119896119894119897 andget the data set of degradation119883119896119894119895Step 7 Simulate the measured degradation 119884119896119894119895 = 119883119896119894119895 + 120576119896119894119895where 120576119896119894119895 sim 119873(0 1205902120576 )53 The Analysis of the Simulated SSADT Data We justgive the analysis of simulated SSADT data here becausewe would give a case study of the CSADT later It wasassumed that the accelerated stress is temperature and thesimulation test contains 4 stresseswhich are 50∘C 60∘C 70∘Cand 80∘C For simplicity the transformed time function isset as Λ(119905) = 119905 and 20 degradation paths are generatedThen the parameters for degradation process are assumedas Θ = 5 4 025 minus3000 001 Moreover the time intervalmeasurement is 25 h for each sample of the whole simulationtest with 40 measurement time points The degradation dataare depicted as in Figure 1

Mathematical Problems in Engineering 9

Table 2 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 log-LF AIC RETruth value 5 1 025 mdash minus3000 001 mdash mdash mdash1198720 501 139 028 mdash minus297561 00078 142228 minus283456 084021198721 512 175 mdash 12907 lowast 10minus4 minus298081 00129 112289 minus223577 562711198722 620 mdash mdash 15213 lowast 10minus4 minus305175 13789 lowast 10minus9 109361 minus217721 78049

0 200 400 600 800 1000

0

05

1

15

2

25

th

y

Figure 1 The simulation degradation paths of SSADT

For simplicity the degradation model for SSADT pro-posed in this paper is referred to as1198720 the model presentedby Tang et al [17] as 1198721 and the stochastic SSADT modeldescribed in reference [13] as 1198722 The estimation results ofthe unknown parameters the log-likelihood function valueand the Akaike information criterion (AIC) [37] which isevaluated by (44) are summarized in Table 2 As we cansee from Table 2 our model clearly outperforms model interms of the log-LF and AIC The estimates of 120583120578 1205902120578 and1205902120576 are more close to the truth values compared with 1198721and 1198722 Meanwhile the diffusion-related variables includeparameters 120581 and 120573 and the diffusion parameter under thefour stresses can be obtained by (10) which is (28377 lowast 10minus548534 lowast 10minus5 78244 lowast 10minus5 12001 lowast 10minus4) in 1198721 and(22334lowast10minus5 39912lowast10minus5 64597lowast10minus5 99423lowast10minus5) forthe true model Moreover taking 1198781 as a reference a criterionformulizing the relative error (RE) is shown as (45) It is clearthat1198720 has the smallest RE from Table 2

AIC (119873) = minus2 ln 119871 (Θ) + 2119873 (44)

RE = 1003816100381610038161003816100381610038161003816100381610038161003816120583120578 minus 120578120578

1003816100381610038161003816100381610038161003816100381610038161003816 +100381610038161003816100381610038161003816100381610038161003816100381610038161205902120578 minus 21205782120578

10038161003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161003816(1205902119861)1 minus (2119861)1(2119861)1

10038161003816100381610038161003816100381610038161003816100381610038161003816+ 1003816100381610038161003816100381610038161003816100381610038161003816120573 minus

1003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161205902120576 minus 21205762120576

1003816100381610038161003816100381610038161003816100381610038161003816 (45)

It is assumed that the failure threshold119908 is equal to 5 andthe normal stress is 25∘C The PDF and CDF under normalstress are shown as in Figure 2 The phenomenon can beobserved where the PDF and CDF of our model most nearlyapproached the real model By contrasting 1198721 with 1198722 itcan be concluded that the neglect of random effects wouldresult in sharper PDF which is consistent with the result inreference [17] And it is noteworthy that the 05 quantiles ofthe failure life are almost equal from the CDF of the threemodels But there is a greater difference between the realmodel and 1198721 while the indicator is taken as mean time tofailure as well as the difference between the real model and1198722 So the correlation between the stress and the diffusionparameter should not be neglected Next we are going toprove the superiority of our model in practical application

6 Case Application

The CSADT model with covariates and random effects isverified by the accelerated degradation data of carbon-filmresistors whose raw data set is explicitly given in Table C3of Meeker and Escobar [38]There were 29 samples and threeaccelerated temperature stresses in the whole test where ninesamples were observed at 83∘C the remaining two stresses133∘C and 173∘C had 10 samples for each stress All of thesamples were observed at the same time points with 1199050 = 01199051 = 452 1199052 = 1030 1199053 = 4341 and 1199054 = 8084 (inhours) Similar to [39] it was assumed that the standardoperating temperature was 50∘C and the threshold value forpercent increase in resistance was taken to be 119862 = 12 Itis reasonable and computationally easier to use the ratios inresistance to the initial value for each rather than the percentincrease [9] For ease of calculation the logarithm of the ratiowas taken as the degradation here That is the value 028of the percent increase was changed to 2796119864 minus 3 and soon and the transformed threshold value 119908 was 01133 Thedegradation paths of all the sample were as shown in Figure 3It can be seen that the degradation of the samples uniformlyshowed a nonlinear characteristic especially at the beginningof the ADT Thus the degradation is modeled with the timescale transformedWiener process as (1) with 120596 = Λ(119905) = 119905119887

Similarly the proposed degradation model for SSADT inthis paper is referred to as1198720 themodel presented by Tang etal [17] as1198721 and themodel described in reference [9] as1198723The estimation results of the unknown parameters the log-likelihood function value and the AIC are shown in Table 3The estimated parameters 119887 are approximately equal for thethree models Take 119887 = 050 as an example to show the time

10 Mathematical Problems in Engineering

Table 3 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 b Log-LF AIC MTTF1198720 823 207 108119864 minus 02 mdash minus420282 182119864 minus 20 050 51848 minus102497 380119864 + 071198721 1471 433 mdash 413119864 minus 07 minus458656 506119864 minus 20 053 48637 minus96073 515119864 + 071198722 1111 mdash mdash 676119864 minus 07 minus447982 411119864 minus 18 053 46837 minus92674 406119864 + 07

0 1 2 3 4 50

05

1

15

2

25

th

The P

DF

Real model

times104

times10minus4

M1

M0

M2

0 1 2 3 4 50

02

04

06

08

1

th

The C

DF

times104

Real modelM1

M0

M2

Figure 2 The comparison of the PDF and CDF of the three models for simulated SSADT data

0 2000 4000 6000 80000

0002

0004

0006

0008

001

th

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 2000 4000 6000 80000

001

002

003

th

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 2000 4000 6000 8000th

0

002

004

006

008

01

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 3 The degradation paths of carbon-film resistors

scale transformed degradation paths as in Figure 4 which areapproximately straight lines compared with Figure 3

Our model has the largest log-LF and smallest AICcompared with1198721 and1198722 The result proves that our modelis more suitable for the degradation data Besides 1198721 hasthe larger log-LF and smaller AIC compared with1198722 So therandom effects could not be neglected It can be observed thatthe variance of the drift parameter is slightly largerwhichmaybe owing to the fixed value of diffusion parameter in1198721

The PDF and CDF under the standard operating temper-ature are as shown in Figure 5The time corresponding to thepeak values of PDF were as follows 2676 lowast 107 1049 lowast 107

and 5028lowast106 for1198720 to1198722 But theMTTF of1198720 isminimalfrom Table 3 And the uncertainty in the estimated PDFs ofthe lifetimes under1198720 is smallest compared to1198721 and1198722 asseen in Figure 5When time 119905 approaches positive infinity theCDF of ourmodel which is calculated by (13) tends illimitablyto 1 and the CDF of 1198721 is roughly equal to 09997 It followsthat our model could do well in distinguishing the differencefrom random effects and the effect of covariates Anotherthing which is worth noting is that the measurement error isquite tiny in the case but it is just an individual phenomenonThemeasurement error should also be considered in practicalapplication

Mathematical Problems in Engineering 11

0 20 40 60 800

0002

0004

0006

0008

001

Λ(t)

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 20 40 60 800

001

002

003

Λ(t)

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 20 40 60 800

002

004

006

008

01

Λ(t)

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 4 The degradation paths of carbon-film resistors under transformed time scale

0 2 4 6 8 100

05

1

15

2

25

3

35

th

times10minus8

M0

M1

M2

The P

DF

times107

0 2 4 6 8 100

02

04

06

08

1

th

M0

M1

M2

The C

DF

times107

Figure 5 The comparison of the PDF and CDF of the three models for real CSADT data

7 Conclusion

In this paper the degradation models based on nonlinearWiener process are established for both constant stress accel-erated degradation data and step stress accelerated degrada-tion data Before the establishment the relationship betweenthe drift parameter and stress variables is derived based onthe invariance principle of failure mechanism and Nelsonassumption so is the relationship between the diffusionparameter and stress variables It is concluded that the ratioof drift parameters under two stresses is a constant which isirrelevant to the testing time and depends only on the twostresses as long as the ratio of diffusion parameters is equalto the ratio of drift parameters And the ratio is defined asaccelerated factor Besides the random effects are also takeninto consideration where the drift parameter is assumed to be

normally distributed and the diffusion parameter is same forall of the samples under a certain stress Then the PDF andCDF of the FHT are deduced considering random effects

Because of the dependency between the diffusion param-eter and stress variables the degradation process is quitedifferent either for CSADT or for SSADT The CSADP andSSADP with random effects are modeled Moreover theunknown parameters are solved by MLE based on the twoproperties of Wiener process At the end of the paper thesimulated data of SSADT and the CSADTdata of carbon-filmresistors are both analyzed to verify the proposed model Itis concluded that the model has the biggest log-LF and thesmallest AIC compared with the two other models

The innovation of this paper lies in the following Firstthe random effects are considered under the new relationshipbetween the diffusion parameter and accelerated stresses

12 Mathematical Problems in Engineering

Second the degradation process was modeled for bothCSADT and SSADT Thirdly the unknown parameters wereestimated based on the two properties of Wiener process andthe result of theMLE for 1205902120578 is discussed on two cases Fourththe measurement error of the degradation data is also valued

However we have only considered the random effects ofthe drift parameter in this paper due to the complexity of thecomputation A further research may consider the randomeffects of the diffusion parameter into the model At the sametime the study of the paper may provide new ideas for therelativity analysis between the parameters of other stochasticprocess and stress variables

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by Jiangsu Province GraduateStudent Scientific Research Innovation Project of China(Project KYLX15 0330)The help is gratefully acknowledged

References

[1] H-F Yu ldquoOptimal classification of highly-reliable productswhose degradation paths satisfyWiener processesrdquo EngineeringOptimization vol 35 no 3 pp 313ndash324 2003

[2] XWang N Balakrishnan and B Guo ldquoResidual life estimationbased on a generalized Wiener degradation processrdquo ReliabilityEngineering and System Safety vol 124 pp 13ndash23 2014

[3] Y Ren Q Feng T Ye and B Sun ldquoA novel model of reliabilityassessment for circular electrical connectorsrdquo IEEE Transac-tions on Components Packaging andManufacturing Technologyvol 5 no 6 pp 755ndash761 2015

[4] Z-S Ye and M Xie ldquoStochastic modelling and analysis ofdegradation for highly reliable productsrdquo Applied StochasticModels in Business and Industry vol 31 no 1 pp 16ndash32 2015

[5] H Lim and B-J Yum ldquoOptimal design of accelerated degrada-tion tests based on Wiener process modelsrdquo Journal of AppliedStatistics vol 38 no 2 pp 309ndash325 2011

[6] C J Lu and W Q Meeker ldquoUsing degradation measures toestimate a time-to-failure distributionrdquo Technometrics vol 35no 2 pp 161ndash174 1993

[7] G AWhitmore ldquoEstimating degradation by a wiener diffusionprocess subject to measurement errorrdquo Lifetime Data Analysisvol 1 no 3 pp 307ndash319 1995

[8] Z Pan and N Balakrishnan ldquoReliability modeling of degra-dation of products with multiple performance characteristicsbased on gamma processesrdquo Reliability Engineering amp SystemSafety vol 96 no 8 pp 949ndash957 2011

[9] C Park and W J Padgett ldquoAccelerated degradation modelsfor failure based on geometric Brownian motion and gammaprocessesrdquo Lifetime Data Analysis vol 11 no 4 pp 511ndash5272005

[10] Z-S Ye and N Chen ldquoThe inverse Gaussian process as adegradation modelrdquo Technometrics vol 56 no 3 pp 302ndash3112014

[11] A Onar and W J Padgett ldquoAccelerated test models with theinverse Gaussian distributionrdquo Journal of Statistical Planningand Inference vol 89 no 1-2 pp 119ndash133 2000

[12] T-R Tsai Y L Lio and N Jiang ldquoOptimal decisions on theaccelerated degradation test plan under the Wiener processrdquoQuality Technology and Quantitative Management vol 11 no4 pp 461ndash470 2014

[13] C-M Liao and S-T Tseng ldquoOptimal design for step-stressaccelerated degradation testsrdquo IEEE Transactions on Reliabilityvol 55 no 1 pp 59ndash66 2006

[14] L C Tang G Yang and M Xie ldquoPlanning of step-stress accel-erated degradation testrdquo in Proceedings of the Annual Reliabil-ity and Maintainability SymposiummdashProceedings InternationalSymposium on Product Quality and Integrity pp 287ndash292January 2004

[15] C-H Hu M-Y Lee and J Tang ldquoOptimum step-stress accel-erated degradation test for Wiener degradation process underconstraintsrdquo European Journal of Operational Research vol 241no 2 pp 412ndash421 2015

[16] S Tang ldquoStep stress accelerated degradation process modelingand remaining useful life estimationrdquo Journal of MechanicalEngineering vol 50 no 16 p 33 2014

[17] S Tang X Guo C Yu H Xue and Z Zhou ldquoAccelerated degra-dation tests modeling based on the nonlinear wiener processwith random effectsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 560726 11 pages 2014

[18] L Wang R Pan X Li and T Jiang ldquoA Bayesian reliability eval-uation method with integrated accelerated degradation testingand field informationrdquo Reliability Engineering amp System Safetyvol 112 pp 38ndash47 2013

[19] Z-S Ye N Chen and Y Shen ldquoA new class of Wiener processmodels for degradation analysisrdquo Reliability Engineering ampSystem Safety vol 139 pp 58ndash67 2015

[20] G A Whitmore and F Schenkelberg ldquoModelling accelerateddegradation data using wiener diffusion with a time scaletransformationrdquo Lifetime Data Analysis vol 3 no 1 pp 27ndash451997

[21] K A Doksum and A Hoyland ldquoModels for variable-stressaccelerated life testing experiments based on Wiener processesand the inverse Gaussian distributionrdquo Technometrics vol 34no 1 pp 74ndash82 1992

[22] H Liao and E A Elsayed ldquoReliability inference for field con-ditions from accelerated degradation testingrdquo Naval ResearchLogistics vol 53 no 6 pp 576ndash587 2006

[23] H-WWang T-X Xu andW-YWang ldquoRemaining life predic-tion based on wiener processes with ADT prior informationrdquoQuality and Reliability Engineering International vol 32 no 3pp 753ndash765 2015

[24] C-Y Peng ldquoInverse Gaussian processes with random effectsand explanatory variables for degradation datardquo Technometricsvol 57 no 1 pp 100ndash111 2015

[25] X Pan XHuang Y Chen YWang and R Kang ldquoConnotationof failure mechanism consistency and identificationmethod foraccelerated testingrdquo in Proceedings of the Prognostics and SystemHealth Management Conference (PHM-Shenzhen rsquo11) May 2011

[26] W B Nelson Accelerated Testing Statistical Models Test Plansand Data Analysis John Wiley amp Sons 2009

[27] Z-S Ye Y Hong and Y Xie ldquoHow do heterogeneities in oper-ating environments affect field failure predictions and testplanningrdquo The Annals of Applied Statistics vol 7 no 4 pp2249ndash2271 2013

Mathematical Problems in Engineering 13

[28] Z-X Zhang X-S Si C-H Hu Q Zhang T Li and C XuldquoPlanning repeated degradation testing for products with three-source variabilityrdquo IEEE Transactions on Reliability vol 65 no2 pp 640ndash647 2016

[29] C-Y Peng and S-T Tseng ldquoMis-specification analysis of lineardegradation modelsrdquo IEEE Transactions on Reliability vol 58no 3 pp 444ndash455 2009

[30] X-S Si W Wang C-H Hu D-H Zhou and M G PechtldquoRemaining useful life estimation based on a nonlinear diffu-sion degradation processrdquo IEEE Transactions on Reliability vol61 no 1 pp 50ndash67 2012

[31] X-S Si W Wang C-H Hu M-Y Chen and D-H Zhou ldquoAWiener-process-based degradationmodel with a recursive filteralgorithm for remaining useful life estimationrdquo MechanicalSystems and Signal Processing vol 35 no 1-2 pp 219ndash237 2013

[32] C-C Tsai S-T Tseng and N Balakrishnan ldquoMis-specificationanalyses of gamma andWiener degradation processesrdquo Journalof Statistical Planning and Inference vol 141 no 12 pp 3725ndash3735 2011

[33] F Haghighi and S J Bae ldquoReliability estimation from lineardegradation and failure time data with competing risks under astep-stress accelerated degradation testrdquo IEEE Transactions onReliability vol 64 no 3 pp 960ndash971 2015

[34] Z-S Ye Y Wang K-L Tsui and M Pecht ldquoDegradation dataanalysis usingwiener processeswithmeasurement errorsrdquo IEEETransactions on Reliability vol 62 no 4 pp 772ndash780 2013

[35] J Huang D S Golubovic S Koh et al ldquoDegradation modelingof mid-power white-light LEDs by using Wiener processrdquoOptics Express vol 23 no 15 pp A966ndashA978 2015

[36] A Beskos O Papaspiliopoulos G O Roberts and P Fearn-head ldquoExact and computationally efficient likelihood-basedestimation for discretely observed diffusion processes (withdiscussion)rdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 68 no 3 pp 333ndash382 2006

[37] J Shang and J E Cavanaugh ldquoAn assumption for the develop-ment of bootstrap variants of the Akaike information criterionin mixed modelsrdquo Statistics and Probability Letters vol 78 no12 pp 1422ndash1429 2008

[38] W QMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[39] W J Padgett andM A Tomlinson ldquoInference from accelerateddegradation and failure data based on Gaussian process mod-elsrdquo Lifetime Data Analysis vol 10 no 2 pp 191ndash206 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

where119872 is themeasurement time point of each samplewhichis the same for all of the samples based on the assumption ofCase 1

The MLE of 120581 120573 and 1205902120576 can be obtained by maximizingthe profile log-likelihood function in (40) through a three-dimensional searchThen substitute them into (37) and (39)we can obtain the MLE for 120583120578 and 1205902120578 Case 2 The number of the measurements and the measure-ment points of each sample are different for all of the samplesunder all of the accelerated stress In this case the first partialderivatives of the log-likelihood function to 1205902120578 are shown asin (35)There may be no analytical form by setting the partialderivatives to zero The evaluation of 120583120578 can be expressed as(36) Then the profile log-likelihood function of 1205902120578 120581 120573 1205902120576can be written as

ln 119871 (120581 120573 120590120576 1205902120578 | 120578 119884)= minus12 ln (2120587) 119873sum

119894=1

119872119894 minus 12 119873sum119894=1

ln (1003816100381610038161003816Π1198941003816100381610038161003816)minus 119873sum119894=1

(Δ119910119894 minus 1205781205891119877119894)Πminus1119894 (Δ119910119894 minus 1205781205891119877119894)1015840 (41)

The MLE of 1205902120578 120581 120573 and 1205902120576 can be obtained by maxi-mizing the profile log-likelihood function in (41) through afour-dimensional search Then substitute them into (36) wecan obtain the MLE for 120583120578

It is not to say that we can only use degradation forCSADT and increment for SSADT but just make an intro-duction to both of the two methods in the limited space Inaddition we could verify the results by comparing the esti-mation calculated by the two methods to avoid computationerrors

5 Simulation Data Analysis

In order to validate the model described before and theparameter estimation methods simulation test was con-ducted

51 The Simulation Method of CSADT Data The parametersΘ = 120583120578 1205902120578 120581 120573 1205902120576 should be given before the simulationand simulation process is shown as in the following steps

Step 1 Set 119896 = 1Step 2 Generate 119873119894 random numbers with subscripts 119894 119894 =1 2 119873119896 subject to normal distribution with mean 120583120578 andvariance 1205902120578 Step 3 Let 119897 = 0Step 4 Calculate119883119896119894(119897+1) using the Euler approximation [36]

119883119896119894(119897+1) = 119883119896119894119897 + 120578119894120589119896120596 (Δ119905) + 120581120589119896radic120596 (Δ119905)Ψ (42)

where Ψ is the random number according to the standardnormal distribution and Δ119905 is the step size in simulation and1198831198961198940 = 0 As such set 119897 = 119897 + 1 and return to Step 4 to get thedegradation path until 119897 = 119871119896119894+1 where 119871119896119894 is the simulationsteps for 119894th under stress 119878119896Step 5 Set 119896 = 119896 + 1 and return to Step 2 until 119896 = 119870 + 1Step 6 Extract the degradation based on the predefinedmeasurement time point 119905119896119894119895 where 119896 = 1 2 119870 119894 =1 2 119873119896 and 119895 = 1 2 119872119896119894 from the data set 119883119896119894119897 andget the data set of degradation119883119896119894119895Step 7 Simulate the measured degradation 119884119896119894119895 = 119883119896119894119895 + 120576119896119894119895where 120576119896119894119895 sim 119873(0 1205902120576 )52 The Simulation Method of SSADT Data There are somedifferences in the simulation process for SSADT comparedwith CSADT The simulation process is shown as follows

Step 1 Generate 119873 random numbers with subscripts 119894 119894 =1 2 119873 subject to normal distribution with mean 120583120578 andvariance 1205902120578 Step 2 Let 119896 = 1Step 3 Set 119897 = 1Step 4 Calculate119883119896119894(119897+1)

119883119896119894(119897+1) = 119883119896119894119897 + 120578119894120589119896120596 (Δ119905) + 120581120589119896radic120596 (Δ119905)Ψ (43)

where Ψ is the random number according to the standardnormal distribution and Δ119905 is the step size in simulation and1198831198961198940 = 01198831198961198940 = 119883(119896minus1)119894119871(119896minus1) (119896 gt 1) As such set 119897 = 119897+1 andreturn to Step 4 to get the degradation path until 119897 = 119871119896119894 + 1Step 5 Set 119896 = 119896 + 1 and return to Step 2 until 119896 = 119870 + 1Step 6 Extract the degradation based on the predefinedmeasurement time point 119905119896119894119895 where 119896 = 1 2 119870 119894 =1 2 119873 and 119895 = 1 2 119872119894119896 from the data set 119883119896119894119897 andget the data set of degradation119883119896119894119895Step 7 Simulate the measured degradation 119884119896119894119895 = 119883119896119894119895 + 120576119896119894119895where 120576119896119894119895 sim 119873(0 1205902120576 )53 The Analysis of the Simulated SSADT Data We justgive the analysis of simulated SSADT data here becausewe would give a case study of the CSADT later It wasassumed that the accelerated stress is temperature and thesimulation test contains 4 stresseswhich are 50∘C 60∘C 70∘Cand 80∘C For simplicity the transformed time function isset as Λ(119905) = 119905 and 20 degradation paths are generatedThen the parameters for degradation process are assumedas Θ = 5 4 025 minus3000 001 Moreover the time intervalmeasurement is 25 h for each sample of the whole simulationtest with 40 measurement time points The degradation dataare depicted as in Figure 1

Mathematical Problems in Engineering 9

Table 2 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 log-LF AIC RETruth value 5 1 025 mdash minus3000 001 mdash mdash mdash1198720 501 139 028 mdash minus297561 00078 142228 minus283456 084021198721 512 175 mdash 12907 lowast 10minus4 minus298081 00129 112289 minus223577 562711198722 620 mdash mdash 15213 lowast 10minus4 minus305175 13789 lowast 10minus9 109361 minus217721 78049

0 200 400 600 800 1000

0

05

1

15

2

25

th

y

Figure 1 The simulation degradation paths of SSADT

For simplicity the degradation model for SSADT pro-posed in this paper is referred to as1198720 the model presentedby Tang et al [17] as 1198721 and the stochastic SSADT modeldescribed in reference [13] as 1198722 The estimation results ofthe unknown parameters the log-likelihood function valueand the Akaike information criterion (AIC) [37] which isevaluated by (44) are summarized in Table 2 As we cansee from Table 2 our model clearly outperforms model interms of the log-LF and AIC The estimates of 120583120578 1205902120578 and1205902120576 are more close to the truth values compared with 1198721and 1198722 Meanwhile the diffusion-related variables includeparameters 120581 and 120573 and the diffusion parameter under thefour stresses can be obtained by (10) which is (28377 lowast 10minus548534 lowast 10minus5 78244 lowast 10minus5 12001 lowast 10minus4) in 1198721 and(22334lowast10minus5 39912lowast10minus5 64597lowast10minus5 99423lowast10minus5) forthe true model Moreover taking 1198781 as a reference a criterionformulizing the relative error (RE) is shown as (45) It is clearthat1198720 has the smallest RE from Table 2

AIC (119873) = minus2 ln 119871 (Θ) + 2119873 (44)

RE = 1003816100381610038161003816100381610038161003816100381610038161003816120583120578 minus 120578120578

1003816100381610038161003816100381610038161003816100381610038161003816 +100381610038161003816100381610038161003816100381610038161003816100381610038161205902120578 minus 21205782120578

10038161003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161003816(1205902119861)1 minus (2119861)1(2119861)1

10038161003816100381610038161003816100381610038161003816100381610038161003816+ 1003816100381610038161003816100381610038161003816100381610038161003816120573 minus

1003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161205902120576 minus 21205762120576

1003816100381610038161003816100381610038161003816100381610038161003816 (45)

It is assumed that the failure threshold119908 is equal to 5 andthe normal stress is 25∘C The PDF and CDF under normalstress are shown as in Figure 2 The phenomenon can beobserved where the PDF and CDF of our model most nearlyapproached the real model By contrasting 1198721 with 1198722 itcan be concluded that the neglect of random effects wouldresult in sharper PDF which is consistent with the result inreference [17] And it is noteworthy that the 05 quantiles ofthe failure life are almost equal from the CDF of the threemodels But there is a greater difference between the realmodel and 1198721 while the indicator is taken as mean time tofailure as well as the difference between the real model and1198722 So the correlation between the stress and the diffusionparameter should not be neglected Next we are going toprove the superiority of our model in practical application

6 Case Application

The CSADT model with covariates and random effects isverified by the accelerated degradation data of carbon-filmresistors whose raw data set is explicitly given in Table C3of Meeker and Escobar [38]There were 29 samples and threeaccelerated temperature stresses in the whole test where ninesamples were observed at 83∘C the remaining two stresses133∘C and 173∘C had 10 samples for each stress All of thesamples were observed at the same time points with 1199050 = 01199051 = 452 1199052 = 1030 1199053 = 4341 and 1199054 = 8084 (inhours) Similar to [39] it was assumed that the standardoperating temperature was 50∘C and the threshold value forpercent increase in resistance was taken to be 119862 = 12 Itis reasonable and computationally easier to use the ratios inresistance to the initial value for each rather than the percentincrease [9] For ease of calculation the logarithm of the ratiowas taken as the degradation here That is the value 028of the percent increase was changed to 2796119864 minus 3 and soon and the transformed threshold value 119908 was 01133 Thedegradation paths of all the sample were as shown in Figure 3It can be seen that the degradation of the samples uniformlyshowed a nonlinear characteristic especially at the beginningof the ADT Thus the degradation is modeled with the timescale transformedWiener process as (1) with 120596 = Λ(119905) = 119905119887

Similarly the proposed degradation model for SSADT inthis paper is referred to as1198720 themodel presented by Tang etal [17] as1198721 and themodel described in reference [9] as1198723The estimation results of the unknown parameters the log-likelihood function value and the AIC are shown in Table 3The estimated parameters 119887 are approximately equal for thethree models Take 119887 = 050 as an example to show the time

10 Mathematical Problems in Engineering

Table 3 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 b Log-LF AIC MTTF1198720 823 207 108119864 minus 02 mdash minus420282 182119864 minus 20 050 51848 minus102497 380119864 + 071198721 1471 433 mdash 413119864 minus 07 minus458656 506119864 minus 20 053 48637 minus96073 515119864 + 071198722 1111 mdash mdash 676119864 minus 07 minus447982 411119864 minus 18 053 46837 minus92674 406119864 + 07

0 1 2 3 4 50

05

1

15

2

25

th

The P

DF

Real model

times104

times10minus4

M1

M0

M2

0 1 2 3 4 50

02

04

06

08

1

th

The C

DF

times104

Real modelM1

M0

M2

Figure 2 The comparison of the PDF and CDF of the three models for simulated SSADT data

0 2000 4000 6000 80000

0002

0004

0006

0008

001

th

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 2000 4000 6000 80000

001

002

003

th

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 2000 4000 6000 8000th

0

002

004

006

008

01

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 3 The degradation paths of carbon-film resistors

scale transformed degradation paths as in Figure 4 which areapproximately straight lines compared with Figure 3

Our model has the largest log-LF and smallest AICcompared with1198721 and1198722 The result proves that our modelis more suitable for the degradation data Besides 1198721 hasthe larger log-LF and smaller AIC compared with1198722 So therandom effects could not be neglected It can be observed thatthe variance of the drift parameter is slightly largerwhichmaybe owing to the fixed value of diffusion parameter in1198721

The PDF and CDF under the standard operating temper-ature are as shown in Figure 5The time corresponding to thepeak values of PDF were as follows 2676 lowast 107 1049 lowast 107

and 5028lowast106 for1198720 to1198722 But theMTTF of1198720 isminimalfrom Table 3 And the uncertainty in the estimated PDFs ofthe lifetimes under1198720 is smallest compared to1198721 and1198722 asseen in Figure 5When time 119905 approaches positive infinity theCDF of ourmodel which is calculated by (13) tends illimitablyto 1 and the CDF of 1198721 is roughly equal to 09997 It followsthat our model could do well in distinguishing the differencefrom random effects and the effect of covariates Anotherthing which is worth noting is that the measurement error isquite tiny in the case but it is just an individual phenomenonThemeasurement error should also be considered in practicalapplication

Mathematical Problems in Engineering 11

0 20 40 60 800

0002

0004

0006

0008

001

Λ(t)

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 20 40 60 800

001

002

003

Λ(t)

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 20 40 60 800

002

004

006

008

01

Λ(t)

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 4 The degradation paths of carbon-film resistors under transformed time scale

0 2 4 6 8 100

05

1

15

2

25

3

35

th

times10minus8

M0

M1

M2

The P

DF

times107

0 2 4 6 8 100

02

04

06

08

1

th

M0

M1

M2

The C

DF

times107

Figure 5 The comparison of the PDF and CDF of the three models for real CSADT data

7 Conclusion

In this paper the degradation models based on nonlinearWiener process are established for both constant stress accel-erated degradation data and step stress accelerated degrada-tion data Before the establishment the relationship betweenthe drift parameter and stress variables is derived based onthe invariance principle of failure mechanism and Nelsonassumption so is the relationship between the diffusionparameter and stress variables It is concluded that the ratioof drift parameters under two stresses is a constant which isirrelevant to the testing time and depends only on the twostresses as long as the ratio of diffusion parameters is equalto the ratio of drift parameters And the ratio is defined asaccelerated factor Besides the random effects are also takeninto consideration where the drift parameter is assumed to be

normally distributed and the diffusion parameter is same forall of the samples under a certain stress Then the PDF andCDF of the FHT are deduced considering random effects

Because of the dependency between the diffusion param-eter and stress variables the degradation process is quitedifferent either for CSADT or for SSADT The CSADP andSSADP with random effects are modeled Moreover theunknown parameters are solved by MLE based on the twoproperties of Wiener process At the end of the paper thesimulated data of SSADT and the CSADTdata of carbon-filmresistors are both analyzed to verify the proposed model Itis concluded that the model has the biggest log-LF and thesmallest AIC compared with the two other models

The innovation of this paper lies in the following Firstthe random effects are considered under the new relationshipbetween the diffusion parameter and accelerated stresses

12 Mathematical Problems in Engineering

Second the degradation process was modeled for bothCSADT and SSADT Thirdly the unknown parameters wereestimated based on the two properties of Wiener process andthe result of theMLE for 1205902120578 is discussed on two cases Fourththe measurement error of the degradation data is also valued

However we have only considered the random effects ofthe drift parameter in this paper due to the complexity of thecomputation A further research may consider the randomeffects of the diffusion parameter into the model At the sametime the study of the paper may provide new ideas for therelativity analysis between the parameters of other stochasticprocess and stress variables

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by Jiangsu Province GraduateStudent Scientific Research Innovation Project of China(Project KYLX15 0330)The help is gratefully acknowledged

References

[1] H-F Yu ldquoOptimal classification of highly-reliable productswhose degradation paths satisfyWiener processesrdquo EngineeringOptimization vol 35 no 3 pp 313ndash324 2003

[2] XWang N Balakrishnan and B Guo ldquoResidual life estimationbased on a generalized Wiener degradation processrdquo ReliabilityEngineering and System Safety vol 124 pp 13ndash23 2014

[3] Y Ren Q Feng T Ye and B Sun ldquoA novel model of reliabilityassessment for circular electrical connectorsrdquo IEEE Transac-tions on Components Packaging andManufacturing Technologyvol 5 no 6 pp 755ndash761 2015

[4] Z-S Ye and M Xie ldquoStochastic modelling and analysis ofdegradation for highly reliable productsrdquo Applied StochasticModels in Business and Industry vol 31 no 1 pp 16ndash32 2015

[5] H Lim and B-J Yum ldquoOptimal design of accelerated degrada-tion tests based on Wiener process modelsrdquo Journal of AppliedStatistics vol 38 no 2 pp 309ndash325 2011

[6] C J Lu and W Q Meeker ldquoUsing degradation measures toestimate a time-to-failure distributionrdquo Technometrics vol 35no 2 pp 161ndash174 1993

[7] G AWhitmore ldquoEstimating degradation by a wiener diffusionprocess subject to measurement errorrdquo Lifetime Data Analysisvol 1 no 3 pp 307ndash319 1995

[8] Z Pan and N Balakrishnan ldquoReliability modeling of degra-dation of products with multiple performance characteristicsbased on gamma processesrdquo Reliability Engineering amp SystemSafety vol 96 no 8 pp 949ndash957 2011

[9] C Park and W J Padgett ldquoAccelerated degradation modelsfor failure based on geometric Brownian motion and gammaprocessesrdquo Lifetime Data Analysis vol 11 no 4 pp 511ndash5272005

[10] Z-S Ye and N Chen ldquoThe inverse Gaussian process as adegradation modelrdquo Technometrics vol 56 no 3 pp 302ndash3112014

[11] A Onar and W J Padgett ldquoAccelerated test models with theinverse Gaussian distributionrdquo Journal of Statistical Planningand Inference vol 89 no 1-2 pp 119ndash133 2000

[12] T-R Tsai Y L Lio and N Jiang ldquoOptimal decisions on theaccelerated degradation test plan under the Wiener processrdquoQuality Technology and Quantitative Management vol 11 no4 pp 461ndash470 2014

[13] C-M Liao and S-T Tseng ldquoOptimal design for step-stressaccelerated degradation testsrdquo IEEE Transactions on Reliabilityvol 55 no 1 pp 59ndash66 2006

[14] L C Tang G Yang and M Xie ldquoPlanning of step-stress accel-erated degradation testrdquo in Proceedings of the Annual Reliabil-ity and Maintainability SymposiummdashProceedings InternationalSymposium on Product Quality and Integrity pp 287ndash292January 2004

[15] C-H Hu M-Y Lee and J Tang ldquoOptimum step-stress accel-erated degradation test for Wiener degradation process underconstraintsrdquo European Journal of Operational Research vol 241no 2 pp 412ndash421 2015

[16] S Tang ldquoStep stress accelerated degradation process modelingand remaining useful life estimationrdquo Journal of MechanicalEngineering vol 50 no 16 p 33 2014

[17] S Tang X Guo C Yu H Xue and Z Zhou ldquoAccelerated degra-dation tests modeling based on the nonlinear wiener processwith random effectsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 560726 11 pages 2014

[18] L Wang R Pan X Li and T Jiang ldquoA Bayesian reliability eval-uation method with integrated accelerated degradation testingand field informationrdquo Reliability Engineering amp System Safetyvol 112 pp 38ndash47 2013

[19] Z-S Ye N Chen and Y Shen ldquoA new class of Wiener processmodels for degradation analysisrdquo Reliability Engineering ampSystem Safety vol 139 pp 58ndash67 2015

[20] G A Whitmore and F Schenkelberg ldquoModelling accelerateddegradation data using wiener diffusion with a time scaletransformationrdquo Lifetime Data Analysis vol 3 no 1 pp 27ndash451997

[21] K A Doksum and A Hoyland ldquoModels for variable-stressaccelerated life testing experiments based on Wiener processesand the inverse Gaussian distributionrdquo Technometrics vol 34no 1 pp 74ndash82 1992

[22] H Liao and E A Elsayed ldquoReliability inference for field con-ditions from accelerated degradation testingrdquo Naval ResearchLogistics vol 53 no 6 pp 576ndash587 2006

[23] H-WWang T-X Xu andW-YWang ldquoRemaining life predic-tion based on wiener processes with ADT prior informationrdquoQuality and Reliability Engineering International vol 32 no 3pp 753ndash765 2015

[24] C-Y Peng ldquoInverse Gaussian processes with random effectsand explanatory variables for degradation datardquo Technometricsvol 57 no 1 pp 100ndash111 2015

[25] X Pan XHuang Y Chen YWang and R Kang ldquoConnotationof failure mechanism consistency and identificationmethod foraccelerated testingrdquo in Proceedings of the Prognostics and SystemHealth Management Conference (PHM-Shenzhen rsquo11) May 2011

[26] W B Nelson Accelerated Testing Statistical Models Test Plansand Data Analysis John Wiley amp Sons 2009

[27] Z-S Ye Y Hong and Y Xie ldquoHow do heterogeneities in oper-ating environments affect field failure predictions and testplanningrdquo The Annals of Applied Statistics vol 7 no 4 pp2249ndash2271 2013

Mathematical Problems in Engineering 13

[28] Z-X Zhang X-S Si C-H Hu Q Zhang T Li and C XuldquoPlanning repeated degradation testing for products with three-source variabilityrdquo IEEE Transactions on Reliability vol 65 no2 pp 640ndash647 2016

[29] C-Y Peng and S-T Tseng ldquoMis-specification analysis of lineardegradation modelsrdquo IEEE Transactions on Reliability vol 58no 3 pp 444ndash455 2009

[30] X-S Si W Wang C-H Hu D-H Zhou and M G PechtldquoRemaining useful life estimation based on a nonlinear diffu-sion degradation processrdquo IEEE Transactions on Reliability vol61 no 1 pp 50ndash67 2012

[31] X-S Si W Wang C-H Hu M-Y Chen and D-H Zhou ldquoAWiener-process-based degradationmodel with a recursive filteralgorithm for remaining useful life estimationrdquo MechanicalSystems and Signal Processing vol 35 no 1-2 pp 219ndash237 2013

[32] C-C Tsai S-T Tseng and N Balakrishnan ldquoMis-specificationanalyses of gamma andWiener degradation processesrdquo Journalof Statistical Planning and Inference vol 141 no 12 pp 3725ndash3735 2011

[33] F Haghighi and S J Bae ldquoReliability estimation from lineardegradation and failure time data with competing risks under astep-stress accelerated degradation testrdquo IEEE Transactions onReliability vol 64 no 3 pp 960ndash971 2015

[34] Z-S Ye Y Wang K-L Tsui and M Pecht ldquoDegradation dataanalysis usingwiener processeswithmeasurement errorsrdquo IEEETransactions on Reliability vol 62 no 4 pp 772ndash780 2013

[35] J Huang D S Golubovic S Koh et al ldquoDegradation modelingof mid-power white-light LEDs by using Wiener processrdquoOptics Express vol 23 no 15 pp A966ndashA978 2015

[36] A Beskos O Papaspiliopoulos G O Roberts and P Fearn-head ldquoExact and computationally efficient likelihood-basedestimation for discretely observed diffusion processes (withdiscussion)rdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 68 no 3 pp 333ndash382 2006

[37] J Shang and J E Cavanaugh ldquoAn assumption for the develop-ment of bootstrap variants of the Akaike information criterionin mixed modelsrdquo Statistics and Probability Letters vol 78 no12 pp 1422ndash1429 2008

[38] W QMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[39] W J Padgett andM A Tomlinson ldquoInference from accelerateddegradation and failure data based on Gaussian process mod-elsrdquo Lifetime Data Analysis vol 10 no 2 pp 191ndash206 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 2 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 log-LF AIC RETruth value 5 1 025 mdash minus3000 001 mdash mdash mdash1198720 501 139 028 mdash minus297561 00078 142228 minus283456 084021198721 512 175 mdash 12907 lowast 10minus4 minus298081 00129 112289 minus223577 562711198722 620 mdash mdash 15213 lowast 10minus4 minus305175 13789 lowast 10minus9 109361 minus217721 78049

0 200 400 600 800 1000

0

05

1

15

2

25

th

y

Figure 1 The simulation degradation paths of SSADT

For simplicity the degradation model for SSADT pro-posed in this paper is referred to as1198720 the model presentedby Tang et al [17] as 1198721 and the stochastic SSADT modeldescribed in reference [13] as 1198722 The estimation results ofthe unknown parameters the log-likelihood function valueand the Akaike information criterion (AIC) [37] which isevaluated by (44) are summarized in Table 2 As we cansee from Table 2 our model clearly outperforms model interms of the log-LF and AIC The estimates of 120583120578 1205902120578 and1205902120576 are more close to the truth values compared with 1198721and 1198722 Meanwhile the diffusion-related variables includeparameters 120581 and 120573 and the diffusion parameter under thefour stresses can be obtained by (10) which is (28377 lowast 10minus548534 lowast 10minus5 78244 lowast 10minus5 12001 lowast 10minus4) in 1198721 and(22334lowast10minus5 39912lowast10minus5 64597lowast10minus5 99423lowast10minus5) forthe true model Moreover taking 1198781 as a reference a criterionformulizing the relative error (RE) is shown as (45) It is clearthat1198720 has the smallest RE from Table 2

AIC (119873) = minus2 ln 119871 (Θ) + 2119873 (44)

RE = 1003816100381610038161003816100381610038161003816100381610038161003816120583120578 minus 120578120578

1003816100381610038161003816100381610038161003816100381610038161003816 +100381610038161003816100381610038161003816100381610038161003816100381610038161205902120578 minus 21205782120578

10038161003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161003816(1205902119861)1 minus (2119861)1(2119861)1

10038161003816100381610038161003816100381610038161003816100381610038161003816+ 1003816100381610038161003816100381610038161003816100381610038161003816120573 minus

1003816100381610038161003816100381610038161003816100381610038161003816 +10038161003816100381610038161003816100381610038161003816100381610038161205902120576 minus 21205762120576

1003816100381610038161003816100381610038161003816100381610038161003816 (45)

It is assumed that the failure threshold119908 is equal to 5 andthe normal stress is 25∘C The PDF and CDF under normalstress are shown as in Figure 2 The phenomenon can beobserved where the PDF and CDF of our model most nearlyapproached the real model By contrasting 1198721 with 1198722 itcan be concluded that the neglect of random effects wouldresult in sharper PDF which is consistent with the result inreference [17] And it is noteworthy that the 05 quantiles ofthe failure life are almost equal from the CDF of the threemodels But there is a greater difference between the realmodel and 1198721 while the indicator is taken as mean time tofailure as well as the difference between the real model and1198722 So the correlation between the stress and the diffusionparameter should not be neglected Next we are going toprove the superiority of our model in practical application

6 Case Application

The CSADT model with covariates and random effects isverified by the accelerated degradation data of carbon-filmresistors whose raw data set is explicitly given in Table C3of Meeker and Escobar [38]There were 29 samples and threeaccelerated temperature stresses in the whole test where ninesamples were observed at 83∘C the remaining two stresses133∘C and 173∘C had 10 samples for each stress All of thesamples were observed at the same time points with 1199050 = 01199051 = 452 1199052 = 1030 1199053 = 4341 and 1199054 = 8084 (inhours) Similar to [39] it was assumed that the standardoperating temperature was 50∘C and the threshold value forpercent increase in resistance was taken to be 119862 = 12 Itis reasonable and computationally easier to use the ratios inresistance to the initial value for each rather than the percentincrease [9] For ease of calculation the logarithm of the ratiowas taken as the degradation here That is the value 028of the percent increase was changed to 2796119864 minus 3 and soon and the transformed threshold value 119908 was 01133 Thedegradation paths of all the sample were as shown in Figure 3It can be seen that the degradation of the samples uniformlyshowed a nonlinear characteristic especially at the beginningof the ADT Thus the degradation is modeled with the timescale transformedWiener process as (1) with 120596 = Λ(119905) = 119905119887

Similarly the proposed degradation model for SSADT inthis paper is referred to as1198720 themodel presented by Tang etal [17] as1198721 and themodel described in reference [9] as1198723The estimation results of the unknown parameters the log-likelihood function value and the AIC are shown in Table 3The estimated parameters 119887 are approximately equal for thethree models Take 119887 = 050 as an example to show the time

10 Mathematical Problems in Engineering

Table 3 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 b Log-LF AIC MTTF1198720 823 207 108119864 minus 02 mdash minus420282 182119864 minus 20 050 51848 minus102497 380119864 + 071198721 1471 433 mdash 413119864 minus 07 minus458656 506119864 minus 20 053 48637 minus96073 515119864 + 071198722 1111 mdash mdash 676119864 minus 07 minus447982 411119864 minus 18 053 46837 minus92674 406119864 + 07

0 1 2 3 4 50

05

1

15

2

25

th

The P

DF

Real model

times104

times10minus4

M1

M0

M2

0 1 2 3 4 50

02

04

06

08

1

th

The C

DF

times104

Real modelM1

M0

M2

Figure 2 The comparison of the PDF and CDF of the three models for simulated SSADT data

0 2000 4000 6000 80000

0002

0004

0006

0008

001

th

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 2000 4000 6000 80000

001

002

003

th

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 2000 4000 6000 8000th

0

002

004

006

008

01

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 3 The degradation paths of carbon-film resistors

scale transformed degradation paths as in Figure 4 which areapproximately straight lines compared with Figure 3

Our model has the largest log-LF and smallest AICcompared with1198721 and1198722 The result proves that our modelis more suitable for the degradation data Besides 1198721 hasthe larger log-LF and smaller AIC compared with1198722 So therandom effects could not be neglected It can be observed thatthe variance of the drift parameter is slightly largerwhichmaybe owing to the fixed value of diffusion parameter in1198721

The PDF and CDF under the standard operating temper-ature are as shown in Figure 5The time corresponding to thepeak values of PDF were as follows 2676 lowast 107 1049 lowast 107

and 5028lowast106 for1198720 to1198722 But theMTTF of1198720 isminimalfrom Table 3 And the uncertainty in the estimated PDFs ofthe lifetimes under1198720 is smallest compared to1198721 and1198722 asseen in Figure 5When time 119905 approaches positive infinity theCDF of ourmodel which is calculated by (13) tends illimitablyto 1 and the CDF of 1198721 is roughly equal to 09997 It followsthat our model could do well in distinguishing the differencefrom random effects and the effect of covariates Anotherthing which is worth noting is that the measurement error isquite tiny in the case but it is just an individual phenomenonThemeasurement error should also be considered in practicalapplication

Mathematical Problems in Engineering 11

0 20 40 60 800

0002

0004

0006

0008

001

Λ(t)

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 20 40 60 800

001

002

003

Λ(t)

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 20 40 60 800

002

004

006

008

01

Λ(t)

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 4 The degradation paths of carbon-film resistors under transformed time scale

0 2 4 6 8 100

05

1

15

2

25

3

35

th

times10minus8

M0

M1

M2

The P

DF

times107

0 2 4 6 8 100

02

04

06

08

1

th

M0

M1

M2

The C

DF

times107

Figure 5 The comparison of the PDF and CDF of the three models for real CSADT data

7 Conclusion

In this paper the degradation models based on nonlinearWiener process are established for both constant stress accel-erated degradation data and step stress accelerated degrada-tion data Before the establishment the relationship betweenthe drift parameter and stress variables is derived based onthe invariance principle of failure mechanism and Nelsonassumption so is the relationship between the diffusionparameter and stress variables It is concluded that the ratioof drift parameters under two stresses is a constant which isirrelevant to the testing time and depends only on the twostresses as long as the ratio of diffusion parameters is equalto the ratio of drift parameters And the ratio is defined asaccelerated factor Besides the random effects are also takeninto consideration where the drift parameter is assumed to be

normally distributed and the diffusion parameter is same forall of the samples under a certain stress Then the PDF andCDF of the FHT are deduced considering random effects

Because of the dependency between the diffusion param-eter and stress variables the degradation process is quitedifferent either for CSADT or for SSADT The CSADP andSSADP with random effects are modeled Moreover theunknown parameters are solved by MLE based on the twoproperties of Wiener process At the end of the paper thesimulated data of SSADT and the CSADTdata of carbon-filmresistors are both analyzed to verify the proposed model Itis concluded that the model has the biggest log-LF and thesmallest AIC compared with the two other models

The innovation of this paper lies in the following Firstthe random effects are considered under the new relationshipbetween the diffusion parameter and accelerated stresses

12 Mathematical Problems in Engineering

Second the degradation process was modeled for bothCSADT and SSADT Thirdly the unknown parameters wereestimated based on the two properties of Wiener process andthe result of theMLE for 1205902120578 is discussed on two cases Fourththe measurement error of the degradation data is also valued

However we have only considered the random effects ofthe drift parameter in this paper due to the complexity of thecomputation A further research may consider the randomeffects of the diffusion parameter into the model At the sametime the study of the paper may provide new ideas for therelativity analysis between the parameters of other stochasticprocess and stress variables

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by Jiangsu Province GraduateStudent Scientific Research Innovation Project of China(Project KYLX15 0330)The help is gratefully acknowledged

References

[1] H-F Yu ldquoOptimal classification of highly-reliable productswhose degradation paths satisfyWiener processesrdquo EngineeringOptimization vol 35 no 3 pp 313ndash324 2003

[2] XWang N Balakrishnan and B Guo ldquoResidual life estimationbased on a generalized Wiener degradation processrdquo ReliabilityEngineering and System Safety vol 124 pp 13ndash23 2014

[3] Y Ren Q Feng T Ye and B Sun ldquoA novel model of reliabilityassessment for circular electrical connectorsrdquo IEEE Transac-tions on Components Packaging andManufacturing Technologyvol 5 no 6 pp 755ndash761 2015

[4] Z-S Ye and M Xie ldquoStochastic modelling and analysis ofdegradation for highly reliable productsrdquo Applied StochasticModels in Business and Industry vol 31 no 1 pp 16ndash32 2015

[5] H Lim and B-J Yum ldquoOptimal design of accelerated degrada-tion tests based on Wiener process modelsrdquo Journal of AppliedStatistics vol 38 no 2 pp 309ndash325 2011

[6] C J Lu and W Q Meeker ldquoUsing degradation measures toestimate a time-to-failure distributionrdquo Technometrics vol 35no 2 pp 161ndash174 1993

[7] G AWhitmore ldquoEstimating degradation by a wiener diffusionprocess subject to measurement errorrdquo Lifetime Data Analysisvol 1 no 3 pp 307ndash319 1995

[8] Z Pan and N Balakrishnan ldquoReliability modeling of degra-dation of products with multiple performance characteristicsbased on gamma processesrdquo Reliability Engineering amp SystemSafety vol 96 no 8 pp 949ndash957 2011

[9] C Park and W J Padgett ldquoAccelerated degradation modelsfor failure based on geometric Brownian motion and gammaprocessesrdquo Lifetime Data Analysis vol 11 no 4 pp 511ndash5272005

[10] Z-S Ye and N Chen ldquoThe inverse Gaussian process as adegradation modelrdquo Technometrics vol 56 no 3 pp 302ndash3112014

[11] A Onar and W J Padgett ldquoAccelerated test models with theinverse Gaussian distributionrdquo Journal of Statistical Planningand Inference vol 89 no 1-2 pp 119ndash133 2000

[12] T-R Tsai Y L Lio and N Jiang ldquoOptimal decisions on theaccelerated degradation test plan under the Wiener processrdquoQuality Technology and Quantitative Management vol 11 no4 pp 461ndash470 2014

[13] C-M Liao and S-T Tseng ldquoOptimal design for step-stressaccelerated degradation testsrdquo IEEE Transactions on Reliabilityvol 55 no 1 pp 59ndash66 2006

[14] L C Tang G Yang and M Xie ldquoPlanning of step-stress accel-erated degradation testrdquo in Proceedings of the Annual Reliabil-ity and Maintainability SymposiummdashProceedings InternationalSymposium on Product Quality and Integrity pp 287ndash292January 2004

[15] C-H Hu M-Y Lee and J Tang ldquoOptimum step-stress accel-erated degradation test for Wiener degradation process underconstraintsrdquo European Journal of Operational Research vol 241no 2 pp 412ndash421 2015

[16] S Tang ldquoStep stress accelerated degradation process modelingand remaining useful life estimationrdquo Journal of MechanicalEngineering vol 50 no 16 p 33 2014

[17] S Tang X Guo C Yu H Xue and Z Zhou ldquoAccelerated degra-dation tests modeling based on the nonlinear wiener processwith random effectsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 560726 11 pages 2014

[18] L Wang R Pan X Li and T Jiang ldquoA Bayesian reliability eval-uation method with integrated accelerated degradation testingand field informationrdquo Reliability Engineering amp System Safetyvol 112 pp 38ndash47 2013

[19] Z-S Ye N Chen and Y Shen ldquoA new class of Wiener processmodels for degradation analysisrdquo Reliability Engineering ampSystem Safety vol 139 pp 58ndash67 2015

[20] G A Whitmore and F Schenkelberg ldquoModelling accelerateddegradation data using wiener diffusion with a time scaletransformationrdquo Lifetime Data Analysis vol 3 no 1 pp 27ndash451997

[21] K A Doksum and A Hoyland ldquoModels for variable-stressaccelerated life testing experiments based on Wiener processesand the inverse Gaussian distributionrdquo Technometrics vol 34no 1 pp 74ndash82 1992

[22] H Liao and E A Elsayed ldquoReliability inference for field con-ditions from accelerated degradation testingrdquo Naval ResearchLogistics vol 53 no 6 pp 576ndash587 2006

[23] H-WWang T-X Xu andW-YWang ldquoRemaining life predic-tion based on wiener processes with ADT prior informationrdquoQuality and Reliability Engineering International vol 32 no 3pp 753ndash765 2015

[24] C-Y Peng ldquoInverse Gaussian processes with random effectsand explanatory variables for degradation datardquo Technometricsvol 57 no 1 pp 100ndash111 2015

[25] X Pan XHuang Y Chen YWang and R Kang ldquoConnotationof failure mechanism consistency and identificationmethod foraccelerated testingrdquo in Proceedings of the Prognostics and SystemHealth Management Conference (PHM-Shenzhen rsquo11) May 2011

[26] W B Nelson Accelerated Testing Statistical Models Test Plansand Data Analysis John Wiley amp Sons 2009

[27] Z-S Ye Y Hong and Y Xie ldquoHow do heterogeneities in oper-ating environments affect field failure predictions and testplanningrdquo The Annals of Applied Statistics vol 7 no 4 pp2249ndash2271 2013

Mathematical Problems in Engineering 13

[28] Z-X Zhang X-S Si C-H Hu Q Zhang T Li and C XuldquoPlanning repeated degradation testing for products with three-source variabilityrdquo IEEE Transactions on Reliability vol 65 no2 pp 640ndash647 2016

[29] C-Y Peng and S-T Tseng ldquoMis-specification analysis of lineardegradation modelsrdquo IEEE Transactions on Reliability vol 58no 3 pp 444ndash455 2009

[30] X-S Si W Wang C-H Hu D-H Zhou and M G PechtldquoRemaining useful life estimation based on a nonlinear diffu-sion degradation processrdquo IEEE Transactions on Reliability vol61 no 1 pp 50ndash67 2012

[31] X-S Si W Wang C-H Hu M-Y Chen and D-H Zhou ldquoAWiener-process-based degradationmodel with a recursive filteralgorithm for remaining useful life estimationrdquo MechanicalSystems and Signal Processing vol 35 no 1-2 pp 219ndash237 2013

[32] C-C Tsai S-T Tseng and N Balakrishnan ldquoMis-specificationanalyses of gamma andWiener degradation processesrdquo Journalof Statistical Planning and Inference vol 141 no 12 pp 3725ndash3735 2011

[33] F Haghighi and S J Bae ldquoReliability estimation from lineardegradation and failure time data with competing risks under astep-stress accelerated degradation testrdquo IEEE Transactions onReliability vol 64 no 3 pp 960ndash971 2015

[34] Z-S Ye Y Wang K-L Tsui and M Pecht ldquoDegradation dataanalysis usingwiener processeswithmeasurement errorsrdquo IEEETransactions on Reliability vol 62 no 4 pp 772ndash780 2013

[35] J Huang D S Golubovic S Koh et al ldquoDegradation modelingof mid-power white-light LEDs by using Wiener processrdquoOptics Express vol 23 no 15 pp A966ndashA978 2015

[36] A Beskos O Papaspiliopoulos G O Roberts and P Fearn-head ldquoExact and computationally efficient likelihood-basedestimation for discretely observed diffusion processes (withdiscussion)rdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 68 no 3 pp 333ndash382 2006

[37] J Shang and J E Cavanaugh ldquoAn assumption for the develop-ment of bootstrap variants of the Akaike information criterionin mixed modelsrdquo Statistics and Probability Letters vol 78 no12 pp 1422ndash1429 2008

[38] W QMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[39] W J Padgett andM A Tomlinson ldquoInference from accelerateddegradation and failure data based on Gaussian process mod-elsrdquo Lifetime Data Analysis vol 10 no 2 pp 191ndash206 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Table 3 The parameters of three degradation models with the SSADT simulated degradation data

120583120578 1205902120578 120581 1205902119861 120573 1205902120576 b Log-LF AIC MTTF1198720 823 207 108119864 minus 02 mdash minus420282 182119864 minus 20 050 51848 minus102497 380119864 + 071198721 1471 433 mdash 413119864 minus 07 minus458656 506119864 minus 20 053 48637 minus96073 515119864 + 071198722 1111 mdash mdash 676119864 minus 07 minus447982 411119864 minus 18 053 46837 minus92674 406119864 + 07

0 1 2 3 4 50

05

1

15

2

25

th

The P

DF

Real model

times104

times10minus4

M1

M0

M2

0 1 2 3 4 50

02

04

06

08

1

th

The C

DF

times104

Real modelM1

M0

M2

Figure 2 The comparison of the PDF and CDF of the three models for simulated SSADT data

0 2000 4000 6000 80000

0002

0004

0006

0008

001

th

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 2000 4000 6000 80000

001

002

003

th

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 2000 4000 6000 8000th

0

002

004

006

008

01

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 3 The degradation paths of carbon-film resistors

scale transformed degradation paths as in Figure 4 which areapproximately straight lines compared with Figure 3

Our model has the largest log-LF and smallest AICcompared with1198721 and1198722 The result proves that our modelis more suitable for the degradation data Besides 1198721 hasthe larger log-LF and smaller AIC compared with1198722 So therandom effects could not be neglected It can be observed thatthe variance of the drift parameter is slightly largerwhichmaybe owing to the fixed value of diffusion parameter in1198721

The PDF and CDF under the standard operating temper-ature are as shown in Figure 5The time corresponding to thepeak values of PDF were as follows 2676 lowast 107 1049 lowast 107

and 5028lowast106 for1198720 to1198722 But theMTTF of1198720 isminimalfrom Table 3 And the uncertainty in the estimated PDFs ofthe lifetimes under1198720 is smallest compared to1198721 and1198722 asseen in Figure 5When time 119905 approaches positive infinity theCDF of ourmodel which is calculated by (13) tends illimitablyto 1 and the CDF of 1198721 is roughly equal to 09997 It followsthat our model could do well in distinguishing the differencefrom random effects and the effect of covariates Anotherthing which is worth noting is that the measurement error isquite tiny in the case but it is just an individual phenomenonThemeasurement error should also be considered in practicalapplication

Mathematical Problems in Engineering 11

0 20 40 60 800

0002

0004

0006

0008

001

Λ(t)

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 20 40 60 800

001

002

003

Λ(t)

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 20 40 60 800

002

004

006

008

01

Λ(t)

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 4 The degradation paths of carbon-film resistors under transformed time scale

0 2 4 6 8 100

05

1

15

2

25

3

35

th

times10minus8

M0

M1

M2

The P

DF

times107

0 2 4 6 8 100

02

04

06

08

1

th

M0

M1

M2

The C

DF

times107

Figure 5 The comparison of the PDF and CDF of the three models for real CSADT data

7 Conclusion

In this paper the degradation models based on nonlinearWiener process are established for both constant stress accel-erated degradation data and step stress accelerated degrada-tion data Before the establishment the relationship betweenthe drift parameter and stress variables is derived based onthe invariance principle of failure mechanism and Nelsonassumption so is the relationship between the diffusionparameter and stress variables It is concluded that the ratioof drift parameters under two stresses is a constant which isirrelevant to the testing time and depends only on the twostresses as long as the ratio of diffusion parameters is equalto the ratio of drift parameters And the ratio is defined asaccelerated factor Besides the random effects are also takeninto consideration where the drift parameter is assumed to be

normally distributed and the diffusion parameter is same forall of the samples under a certain stress Then the PDF andCDF of the FHT are deduced considering random effects

Because of the dependency between the diffusion param-eter and stress variables the degradation process is quitedifferent either for CSADT or for SSADT The CSADP andSSADP with random effects are modeled Moreover theunknown parameters are solved by MLE based on the twoproperties of Wiener process At the end of the paper thesimulated data of SSADT and the CSADTdata of carbon-filmresistors are both analyzed to verify the proposed model Itis concluded that the model has the biggest log-LF and thesmallest AIC compared with the two other models

The innovation of this paper lies in the following Firstthe random effects are considered under the new relationshipbetween the diffusion parameter and accelerated stresses

12 Mathematical Problems in Engineering

Second the degradation process was modeled for bothCSADT and SSADT Thirdly the unknown parameters wereestimated based on the two properties of Wiener process andthe result of theMLE for 1205902120578 is discussed on two cases Fourththe measurement error of the degradation data is also valued

However we have only considered the random effects ofthe drift parameter in this paper due to the complexity of thecomputation A further research may consider the randomeffects of the diffusion parameter into the model At the sametime the study of the paper may provide new ideas for therelativity analysis between the parameters of other stochasticprocess and stress variables

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by Jiangsu Province GraduateStudent Scientific Research Innovation Project of China(Project KYLX15 0330)The help is gratefully acknowledged

References

[1] H-F Yu ldquoOptimal classification of highly-reliable productswhose degradation paths satisfyWiener processesrdquo EngineeringOptimization vol 35 no 3 pp 313ndash324 2003

[2] XWang N Balakrishnan and B Guo ldquoResidual life estimationbased on a generalized Wiener degradation processrdquo ReliabilityEngineering and System Safety vol 124 pp 13ndash23 2014

[3] Y Ren Q Feng T Ye and B Sun ldquoA novel model of reliabilityassessment for circular electrical connectorsrdquo IEEE Transac-tions on Components Packaging andManufacturing Technologyvol 5 no 6 pp 755ndash761 2015

[4] Z-S Ye and M Xie ldquoStochastic modelling and analysis ofdegradation for highly reliable productsrdquo Applied StochasticModels in Business and Industry vol 31 no 1 pp 16ndash32 2015

[5] H Lim and B-J Yum ldquoOptimal design of accelerated degrada-tion tests based on Wiener process modelsrdquo Journal of AppliedStatistics vol 38 no 2 pp 309ndash325 2011

[6] C J Lu and W Q Meeker ldquoUsing degradation measures toestimate a time-to-failure distributionrdquo Technometrics vol 35no 2 pp 161ndash174 1993

[7] G AWhitmore ldquoEstimating degradation by a wiener diffusionprocess subject to measurement errorrdquo Lifetime Data Analysisvol 1 no 3 pp 307ndash319 1995

[8] Z Pan and N Balakrishnan ldquoReliability modeling of degra-dation of products with multiple performance characteristicsbased on gamma processesrdquo Reliability Engineering amp SystemSafety vol 96 no 8 pp 949ndash957 2011

[9] C Park and W J Padgett ldquoAccelerated degradation modelsfor failure based on geometric Brownian motion and gammaprocessesrdquo Lifetime Data Analysis vol 11 no 4 pp 511ndash5272005

[10] Z-S Ye and N Chen ldquoThe inverse Gaussian process as adegradation modelrdquo Technometrics vol 56 no 3 pp 302ndash3112014

[11] A Onar and W J Padgett ldquoAccelerated test models with theinverse Gaussian distributionrdquo Journal of Statistical Planningand Inference vol 89 no 1-2 pp 119ndash133 2000

[12] T-R Tsai Y L Lio and N Jiang ldquoOptimal decisions on theaccelerated degradation test plan under the Wiener processrdquoQuality Technology and Quantitative Management vol 11 no4 pp 461ndash470 2014

[13] C-M Liao and S-T Tseng ldquoOptimal design for step-stressaccelerated degradation testsrdquo IEEE Transactions on Reliabilityvol 55 no 1 pp 59ndash66 2006

[14] L C Tang G Yang and M Xie ldquoPlanning of step-stress accel-erated degradation testrdquo in Proceedings of the Annual Reliabil-ity and Maintainability SymposiummdashProceedings InternationalSymposium on Product Quality and Integrity pp 287ndash292January 2004

[15] C-H Hu M-Y Lee and J Tang ldquoOptimum step-stress accel-erated degradation test for Wiener degradation process underconstraintsrdquo European Journal of Operational Research vol 241no 2 pp 412ndash421 2015

[16] S Tang ldquoStep stress accelerated degradation process modelingand remaining useful life estimationrdquo Journal of MechanicalEngineering vol 50 no 16 p 33 2014

[17] S Tang X Guo C Yu H Xue and Z Zhou ldquoAccelerated degra-dation tests modeling based on the nonlinear wiener processwith random effectsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 560726 11 pages 2014

[18] L Wang R Pan X Li and T Jiang ldquoA Bayesian reliability eval-uation method with integrated accelerated degradation testingand field informationrdquo Reliability Engineering amp System Safetyvol 112 pp 38ndash47 2013

[19] Z-S Ye N Chen and Y Shen ldquoA new class of Wiener processmodels for degradation analysisrdquo Reliability Engineering ampSystem Safety vol 139 pp 58ndash67 2015

[20] G A Whitmore and F Schenkelberg ldquoModelling accelerateddegradation data using wiener diffusion with a time scaletransformationrdquo Lifetime Data Analysis vol 3 no 1 pp 27ndash451997

[21] K A Doksum and A Hoyland ldquoModels for variable-stressaccelerated life testing experiments based on Wiener processesand the inverse Gaussian distributionrdquo Technometrics vol 34no 1 pp 74ndash82 1992

[22] H Liao and E A Elsayed ldquoReliability inference for field con-ditions from accelerated degradation testingrdquo Naval ResearchLogistics vol 53 no 6 pp 576ndash587 2006

[23] H-WWang T-X Xu andW-YWang ldquoRemaining life predic-tion based on wiener processes with ADT prior informationrdquoQuality and Reliability Engineering International vol 32 no 3pp 753ndash765 2015

[24] C-Y Peng ldquoInverse Gaussian processes with random effectsand explanatory variables for degradation datardquo Technometricsvol 57 no 1 pp 100ndash111 2015

[25] X Pan XHuang Y Chen YWang and R Kang ldquoConnotationof failure mechanism consistency and identificationmethod foraccelerated testingrdquo in Proceedings of the Prognostics and SystemHealth Management Conference (PHM-Shenzhen rsquo11) May 2011

[26] W B Nelson Accelerated Testing Statistical Models Test Plansand Data Analysis John Wiley amp Sons 2009

[27] Z-S Ye Y Hong and Y Xie ldquoHow do heterogeneities in oper-ating environments affect field failure predictions and testplanningrdquo The Annals of Applied Statistics vol 7 no 4 pp2249ndash2271 2013

Mathematical Problems in Engineering 13

[28] Z-X Zhang X-S Si C-H Hu Q Zhang T Li and C XuldquoPlanning repeated degradation testing for products with three-source variabilityrdquo IEEE Transactions on Reliability vol 65 no2 pp 640ndash647 2016

[29] C-Y Peng and S-T Tseng ldquoMis-specification analysis of lineardegradation modelsrdquo IEEE Transactions on Reliability vol 58no 3 pp 444ndash455 2009

[30] X-S Si W Wang C-H Hu D-H Zhou and M G PechtldquoRemaining useful life estimation based on a nonlinear diffu-sion degradation processrdquo IEEE Transactions on Reliability vol61 no 1 pp 50ndash67 2012

[31] X-S Si W Wang C-H Hu M-Y Chen and D-H Zhou ldquoAWiener-process-based degradationmodel with a recursive filteralgorithm for remaining useful life estimationrdquo MechanicalSystems and Signal Processing vol 35 no 1-2 pp 219ndash237 2013

[32] C-C Tsai S-T Tseng and N Balakrishnan ldquoMis-specificationanalyses of gamma andWiener degradation processesrdquo Journalof Statistical Planning and Inference vol 141 no 12 pp 3725ndash3735 2011

[33] F Haghighi and S J Bae ldquoReliability estimation from lineardegradation and failure time data with competing risks under astep-stress accelerated degradation testrdquo IEEE Transactions onReliability vol 64 no 3 pp 960ndash971 2015

[34] Z-S Ye Y Wang K-L Tsui and M Pecht ldquoDegradation dataanalysis usingwiener processeswithmeasurement errorsrdquo IEEETransactions on Reliability vol 62 no 4 pp 772ndash780 2013

[35] J Huang D S Golubovic S Koh et al ldquoDegradation modelingof mid-power white-light LEDs by using Wiener processrdquoOptics Express vol 23 no 15 pp A966ndashA978 2015

[36] A Beskos O Papaspiliopoulos G O Roberts and P Fearn-head ldquoExact and computationally efficient likelihood-basedestimation for discretely observed diffusion processes (withdiscussion)rdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 68 no 3 pp 333ndash382 2006

[37] J Shang and J E Cavanaugh ldquoAn assumption for the develop-ment of bootstrap variants of the Akaike information criterionin mixed modelsrdquo Statistics and Probability Letters vol 78 no12 pp 1422ndash1429 2008

[38] W QMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[39] W J Padgett andM A Tomlinson ldquoInference from accelerateddegradation and failure data based on Gaussian process mod-elsrdquo Lifetime Data Analysis vol 10 no 2 pp 191ndash206 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0 20 40 60 800

0002

0004

0006

0008

001

Λ(t)

ln(1

minuspercentincrea

se10

0)

(a) 119878 = 83∘C

0 20 40 60 800

001

002

003

Λ(t)

ln(1

minuspercentincrea

se10

0)

(b) 119878 = 133∘C

0 20 40 60 800

002

004

006

008

01

Λ(t)

ln(1

minuspercentincrea

se10

0)

(c) 119878 = 173∘C

Figure 4 The degradation paths of carbon-film resistors under transformed time scale

0 2 4 6 8 100

05

1

15

2

25

3

35

th

times10minus8

M0

M1

M2

The P

DF

times107

0 2 4 6 8 100

02

04

06

08

1

th

M0

M1

M2

The C

DF

times107

Figure 5 The comparison of the PDF and CDF of the three models for real CSADT data

7 Conclusion

In this paper the degradation models based on nonlinearWiener process are established for both constant stress accel-erated degradation data and step stress accelerated degrada-tion data Before the establishment the relationship betweenthe drift parameter and stress variables is derived based onthe invariance principle of failure mechanism and Nelsonassumption so is the relationship between the diffusionparameter and stress variables It is concluded that the ratioof drift parameters under two stresses is a constant which isirrelevant to the testing time and depends only on the twostresses as long as the ratio of diffusion parameters is equalto the ratio of drift parameters And the ratio is defined asaccelerated factor Besides the random effects are also takeninto consideration where the drift parameter is assumed to be

normally distributed and the diffusion parameter is same forall of the samples under a certain stress Then the PDF andCDF of the FHT are deduced considering random effects

Because of the dependency between the diffusion param-eter and stress variables the degradation process is quitedifferent either for CSADT or for SSADT The CSADP andSSADP with random effects are modeled Moreover theunknown parameters are solved by MLE based on the twoproperties of Wiener process At the end of the paper thesimulated data of SSADT and the CSADTdata of carbon-filmresistors are both analyzed to verify the proposed model Itis concluded that the model has the biggest log-LF and thesmallest AIC compared with the two other models

The innovation of this paper lies in the following Firstthe random effects are considered under the new relationshipbetween the diffusion parameter and accelerated stresses

12 Mathematical Problems in Engineering

Second the degradation process was modeled for bothCSADT and SSADT Thirdly the unknown parameters wereestimated based on the two properties of Wiener process andthe result of theMLE for 1205902120578 is discussed on two cases Fourththe measurement error of the degradation data is also valued

However we have only considered the random effects ofthe drift parameter in this paper due to the complexity of thecomputation A further research may consider the randomeffects of the diffusion parameter into the model At the sametime the study of the paper may provide new ideas for therelativity analysis between the parameters of other stochasticprocess and stress variables

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by Jiangsu Province GraduateStudent Scientific Research Innovation Project of China(Project KYLX15 0330)The help is gratefully acknowledged

References

[1] H-F Yu ldquoOptimal classification of highly-reliable productswhose degradation paths satisfyWiener processesrdquo EngineeringOptimization vol 35 no 3 pp 313ndash324 2003

[2] XWang N Balakrishnan and B Guo ldquoResidual life estimationbased on a generalized Wiener degradation processrdquo ReliabilityEngineering and System Safety vol 124 pp 13ndash23 2014

[3] Y Ren Q Feng T Ye and B Sun ldquoA novel model of reliabilityassessment for circular electrical connectorsrdquo IEEE Transac-tions on Components Packaging andManufacturing Technologyvol 5 no 6 pp 755ndash761 2015

[4] Z-S Ye and M Xie ldquoStochastic modelling and analysis ofdegradation for highly reliable productsrdquo Applied StochasticModels in Business and Industry vol 31 no 1 pp 16ndash32 2015

[5] H Lim and B-J Yum ldquoOptimal design of accelerated degrada-tion tests based on Wiener process modelsrdquo Journal of AppliedStatistics vol 38 no 2 pp 309ndash325 2011

[6] C J Lu and W Q Meeker ldquoUsing degradation measures toestimate a time-to-failure distributionrdquo Technometrics vol 35no 2 pp 161ndash174 1993

[7] G AWhitmore ldquoEstimating degradation by a wiener diffusionprocess subject to measurement errorrdquo Lifetime Data Analysisvol 1 no 3 pp 307ndash319 1995

[8] Z Pan and N Balakrishnan ldquoReliability modeling of degra-dation of products with multiple performance characteristicsbased on gamma processesrdquo Reliability Engineering amp SystemSafety vol 96 no 8 pp 949ndash957 2011

[9] C Park and W J Padgett ldquoAccelerated degradation modelsfor failure based on geometric Brownian motion and gammaprocessesrdquo Lifetime Data Analysis vol 11 no 4 pp 511ndash5272005

[10] Z-S Ye and N Chen ldquoThe inverse Gaussian process as adegradation modelrdquo Technometrics vol 56 no 3 pp 302ndash3112014

[11] A Onar and W J Padgett ldquoAccelerated test models with theinverse Gaussian distributionrdquo Journal of Statistical Planningand Inference vol 89 no 1-2 pp 119ndash133 2000

[12] T-R Tsai Y L Lio and N Jiang ldquoOptimal decisions on theaccelerated degradation test plan under the Wiener processrdquoQuality Technology and Quantitative Management vol 11 no4 pp 461ndash470 2014

[13] C-M Liao and S-T Tseng ldquoOptimal design for step-stressaccelerated degradation testsrdquo IEEE Transactions on Reliabilityvol 55 no 1 pp 59ndash66 2006

[14] L C Tang G Yang and M Xie ldquoPlanning of step-stress accel-erated degradation testrdquo in Proceedings of the Annual Reliabil-ity and Maintainability SymposiummdashProceedings InternationalSymposium on Product Quality and Integrity pp 287ndash292January 2004

[15] C-H Hu M-Y Lee and J Tang ldquoOptimum step-stress accel-erated degradation test for Wiener degradation process underconstraintsrdquo European Journal of Operational Research vol 241no 2 pp 412ndash421 2015

[16] S Tang ldquoStep stress accelerated degradation process modelingand remaining useful life estimationrdquo Journal of MechanicalEngineering vol 50 no 16 p 33 2014

[17] S Tang X Guo C Yu H Xue and Z Zhou ldquoAccelerated degra-dation tests modeling based on the nonlinear wiener processwith random effectsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 560726 11 pages 2014

[18] L Wang R Pan X Li and T Jiang ldquoA Bayesian reliability eval-uation method with integrated accelerated degradation testingand field informationrdquo Reliability Engineering amp System Safetyvol 112 pp 38ndash47 2013

[19] Z-S Ye N Chen and Y Shen ldquoA new class of Wiener processmodels for degradation analysisrdquo Reliability Engineering ampSystem Safety vol 139 pp 58ndash67 2015

[20] G A Whitmore and F Schenkelberg ldquoModelling accelerateddegradation data using wiener diffusion with a time scaletransformationrdquo Lifetime Data Analysis vol 3 no 1 pp 27ndash451997

[21] K A Doksum and A Hoyland ldquoModels for variable-stressaccelerated life testing experiments based on Wiener processesand the inverse Gaussian distributionrdquo Technometrics vol 34no 1 pp 74ndash82 1992

[22] H Liao and E A Elsayed ldquoReliability inference for field con-ditions from accelerated degradation testingrdquo Naval ResearchLogistics vol 53 no 6 pp 576ndash587 2006

[23] H-WWang T-X Xu andW-YWang ldquoRemaining life predic-tion based on wiener processes with ADT prior informationrdquoQuality and Reliability Engineering International vol 32 no 3pp 753ndash765 2015

[24] C-Y Peng ldquoInverse Gaussian processes with random effectsand explanatory variables for degradation datardquo Technometricsvol 57 no 1 pp 100ndash111 2015

[25] X Pan XHuang Y Chen YWang and R Kang ldquoConnotationof failure mechanism consistency and identificationmethod foraccelerated testingrdquo in Proceedings of the Prognostics and SystemHealth Management Conference (PHM-Shenzhen rsquo11) May 2011

[26] W B Nelson Accelerated Testing Statistical Models Test Plansand Data Analysis John Wiley amp Sons 2009

[27] Z-S Ye Y Hong and Y Xie ldquoHow do heterogeneities in oper-ating environments affect field failure predictions and testplanningrdquo The Annals of Applied Statistics vol 7 no 4 pp2249ndash2271 2013

Mathematical Problems in Engineering 13

[28] Z-X Zhang X-S Si C-H Hu Q Zhang T Li and C XuldquoPlanning repeated degradation testing for products with three-source variabilityrdquo IEEE Transactions on Reliability vol 65 no2 pp 640ndash647 2016

[29] C-Y Peng and S-T Tseng ldquoMis-specification analysis of lineardegradation modelsrdquo IEEE Transactions on Reliability vol 58no 3 pp 444ndash455 2009

[30] X-S Si W Wang C-H Hu D-H Zhou and M G PechtldquoRemaining useful life estimation based on a nonlinear diffu-sion degradation processrdquo IEEE Transactions on Reliability vol61 no 1 pp 50ndash67 2012

[31] X-S Si W Wang C-H Hu M-Y Chen and D-H Zhou ldquoAWiener-process-based degradationmodel with a recursive filteralgorithm for remaining useful life estimationrdquo MechanicalSystems and Signal Processing vol 35 no 1-2 pp 219ndash237 2013

[32] C-C Tsai S-T Tseng and N Balakrishnan ldquoMis-specificationanalyses of gamma andWiener degradation processesrdquo Journalof Statistical Planning and Inference vol 141 no 12 pp 3725ndash3735 2011

[33] F Haghighi and S J Bae ldquoReliability estimation from lineardegradation and failure time data with competing risks under astep-stress accelerated degradation testrdquo IEEE Transactions onReliability vol 64 no 3 pp 960ndash971 2015

[34] Z-S Ye Y Wang K-L Tsui and M Pecht ldquoDegradation dataanalysis usingwiener processeswithmeasurement errorsrdquo IEEETransactions on Reliability vol 62 no 4 pp 772ndash780 2013

[35] J Huang D S Golubovic S Koh et al ldquoDegradation modelingof mid-power white-light LEDs by using Wiener processrdquoOptics Express vol 23 no 15 pp A966ndashA978 2015

[36] A Beskos O Papaspiliopoulos G O Roberts and P Fearn-head ldquoExact and computationally efficient likelihood-basedestimation for discretely observed diffusion processes (withdiscussion)rdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 68 no 3 pp 333ndash382 2006

[37] J Shang and J E Cavanaugh ldquoAn assumption for the develop-ment of bootstrap variants of the Akaike information criterionin mixed modelsrdquo Statistics and Probability Letters vol 78 no12 pp 1422ndash1429 2008

[38] W QMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[39] W J Padgett andM A Tomlinson ldquoInference from accelerateddegradation and failure data based on Gaussian process mod-elsrdquo Lifetime Data Analysis vol 10 no 2 pp 191ndash206 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

Second the degradation process was modeled for bothCSADT and SSADT Thirdly the unknown parameters wereestimated based on the two properties of Wiener process andthe result of theMLE for 1205902120578 is discussed on two cases Fourththe measurement error of the degradation data is also valued

However we have only considered the random effects ofthe drift parameter in this paper due to the complexity of thecomputation A further research may consider the randomeffects of the diffusion parameter into the model At the sametime the study of the paper may provide new ideas for therelativity analysis between the parameters of other stochasticprocess and stress variables

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by Jiangsu Province GraduateStudent Scientific Research Innovation Project of China(Project KYLX15 0330)The help is gratefully acknowledged

References

[1] H-F Yu ldquoOptimal classification of highly-reliable productswhose degradation paths satisfyWiener processesrdquo EngineeringOptimization vol 35 no 3 pp 313ndash324 2003

[2] XWang N Balakrishnan and B Guo ldquoResidual life estimationbased on a generalized Wiener degradation processrdquo ReliabilityEngineering and System Safety vol 124 pp 13ndash23 2014

[3] Y Ren Q Feng T Ye and B Sun ldquoA novel model of reliabilityassessment for circular electrical connectorsrdquo IEEE Transac-tions on Components Packaging andManufacturing Technologyvol 5 no 6 pp 755ndash761 2015

[4] Z-S Ye and M Xie ldquoStochastic modelling and analysis ofdegradation for highly reliable productsrdquo Applied StochasticModels in Business and Industry vol 31 no 1 pp 16ndash32 2015

[5] H Lim and B-J Yum ldquoOptimal design of accelerated degrada-tion tests based on Wiener process modelsrdquo Journal of AppliedStatistics vol 38 no 2 pp 309ndash325 2011

[6] C J Lu and W Q Meeker ldquoUsing degradation measures toestimate a time-to-failure distributionrdquo Technometrics vol 35no 2 pp 161ndash174 1993

[7] G AWhitmore ldquoEstimating degradation by a wiener diffusionprocess subject to measurement errorrdquo Lifetime Data Analysisvol 1 no 3 pp 307ndash319 1995

[8] Z Pan and N Balakrishnan ldquoReliability modeling of degra-dation of products with multiple performance characteristicsbased on gamma processesrdquo Reliability Engineering amp SystemSafety vol 96 no 8 pp 949ndash957 2011

[9] C Park and W J Padgett ldquoAccelerated degradation modelsfor failure based on geometric Brownian motion and gammaprocessesrdquo Lifetime Data Analysis vol 11 no 4 pp 511ndash5272005

[10] Z-S Ye and N Chen ldquoThe inverse Gaussian process as adegradation modelrdquo Technometrics vol 56 no 3 pp 302ndash3112014

[11] A Onar and W J Padgett ldquoAccelerated test models with theinverse Gaussian distributionrdquo Journal of Statistical Planningand Inference vol 89 no 1-2 pp 119ndash133 2000

[12] T-R Tsai Y L Lio and N Jiang ldquoOptimal decisions on theaccelerated degradation test plan under the Wiener processrdquoQuality Technology and Quantitative Management vol 11 no4 pp 461ndash470 2014

[13] C-M Liao and S-T Tseng ldquoOptimal design for step-stressaccelerated degradation testsrdquo IEEE Transactions on Reliabilityvol 55 no 1 pp 59ndash66 2006

[14] L C Tang G Yang and M Xie ldquoPlanning of step-stress accel-erated degradation testrdquo in Proceedings of the Annual Reliabil-ity and Maintainability SymposiummdashProceedings InternationalSymposium on Product Quality and Integrity pp 287ndash292January 2004

[15] C-H Hu M-Y Lee and J Tang ldquoOptimum step-stress accel-erated degradation test for Wiener degradation process underconstraintsrdquo European Journal of Operational Research vol 241no 2 pp 412ndash421 2015

[16] S Tang ldquoStep stress accelerated degradation process modelingand remaining useful life estimationrdquo Journal of MechanicalEngineering vol 50 no 16 p 33 2014

[17] S Tang X Guo C Yu H Xue and Z Zhou ldquoAccelerated degra-dation tests modeling based on the nonlinear wiener processwith random effectsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 560726 11 pages 2014

[18] L Wang R Pan X Li and T Jiang ldquoA Bayesian reliability eval-uation method with integrated accelerated degradation testingand field informationrdquo Reliability Engineering amp System Safetyvol 112 pp 38ndash47 2013

[19] Z-S Ye N Chen and Y Shen ldquoA new class of Wiener processmodels for degradation analysisrdquo Reliability Engineering ampSystem Safety vol 139 pp 58ndash67 2015

[20] G A Whitmore and F Schenkelberg ldquoModelling accelerateddegradation data using wiener diffusion with a time scaletransformationrdquo Lifetime Data Analysis vol 3 no 1 pp 27ndash451997

[21] K A Doksum and A Hoyland ldquoModels for variable-stressaccelerated life testing experiments based on Wiener processesand the inverse Gaussian distributionrdquo Technometrics vol 34no 1 pp 74ndash82 1992

[22] H Liao and E A Elsayed ldquoReliability inference for field con-ditions from accelerated degradation testingrdquo Naval ResearchLogistics vol 53 no 6 pp 576ndash587 2006

[23] H-WWang T-X Xu andW-YWang ldquoRemaining life predic-tion based on wiener processes with ADT prior informationrdquoQuality and Reliability Engineering International vol 32 no 3pp 753ndash765 2015

[24] C-Y Peng ldquoInverse Gaussian processes with random effectsand explanatory variables for degradation datardquo Technometricsvol 57 no 1 pp 100ndash111 2015

[25] X Pan XHuang Y Chen YWang and R Kang ldquoConnotationof failure mechanism consistency and identificationmethod foraccelerated testingrdquo in Proceedings of the Prognostics and SystemHealth Management Conference (PHM-Shenzhen rsquo11) May 2011

[26] W B Nelson Accelerated Testing Statistical Models Test Plansand Data Analysis John Wiley amp Sons 2009

[27] Z-S Ye Y Hong and Y Xie ldquoHow do heterogeneities in oper-ating environments affect field failure predictions and testplanningrdquo The Annals of Applied Statistics vol 7 no 4 pp2249ndash2271 2013

Mathematical Problems in Engineering 13

[28] Z-X Zhang X-S Si C-H Hu Q Zhang T Li and C XuldquoPlanning repeated degradation testing for products with three-source variabilityrdquo IEEE Transactions on Reliability vol 65 no2 pp 640ndash647 2016

[29] C-Y Peng and S-T Tseng ldquoMis-specification analysis of lineardegradation modelsrdquo IEEE Transactions on Reliability vol 58no 3 pp 444ndash455 2009

[30] X-S Si W Wang C-H Hu D-H Zhou and M G PechtldquoRemaining useful life estimation based on a nonlinear diffu-sion degradation processrdquo IEEE Transactions on Reliability vol61 no 1 pp 50ndash67 2012

[31] X-S Si W Wang C-H Hu M-Y Chen and D-H Zhou ldquoAWiener-process-based degradationmodel with a recursive filteralgorithm for remaining useful life estimationrdquo MechanicalSystems and Signal Processing vol 35 no 1-2 pp 219ndash237 2013

[32] C-C Tsai S-T Tseng and N Balakrishnan ldquoMis-specificationanalyses of gamma andWiener degradation processesrdquo Journalof Statistical Planning and Inference vol 141 no 12 pp 3725ndash3735 2011

[33] F Haghighi and S J Bae ldquoReliability estimation from lineardegradation and failure time data with competing risks under astep-stress accelerated degradation testrdquo IEEE Transactions onReliability vol 64 no 3 pp 960ndash971 2015

[34] Z-S Ye Y Wang K-L Tsui and M Pecht ldquoDegradation dataanalysis usingwiener processeswithmeasurement errorsrdquo IEEETransactions on Reliability vol 62 no 4 pp 772ndash780 2013

[35] J Huang D S Golubovic S Koh et al ldquoDegradation modelingof mid-power white-light LEDs by using Wiener processrdquoOptics Express vol 23 no 15 pp A966ndashA978 2015

[36] A Beskos O Papaspiliopoulos G O Roberts and P Fearn-head ldquoExact and computationally efficient likelihood-basedestimation for discretely observed diffusion processes (withdiscussion)rdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 68 no 3 pp 333ndash382 2006

[37] J Shang and J E Cavanaugh ldquoAn assumption for the develop-ment of bootstrap variants of the Akaike information criterionin mixed modelsrdquo Statistics and Probability Letters vol 78 no12 pp 1422ndash1429 2008

[38] W QMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[39] W J Padgett andM A Tomlinson ldquoInference from accelerateddegradation and failure data based on Gaussian process mod-elsrdquo Lifetime Data Analysis vol 10 no 2 pp 191ndash206 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

[28] Z-X Zhang X-S Si C-H Hu Q Zhang T Li and C XuldquoPlanning repeated degradation testing for products with three-source variabilityrdquo IEEE Transactions on Reliability vol 65 no2 pp 640ndash647 2016

[29] C-Y Peng and S-T Tseng ldquoMis-specification analysis of lineardegradation modelsrdquo IEEE Transactions on Reliability vol 58no 3 pp 444ndash455 2009

[30] X-S Si W Wang C-H Hu D-H Zhou and M G PechtldquoRemaining useful life estimation based on a nonlinear diffu-sion degradation processrdquo IEEE Transactions on Reliability vol61 no 1 pp 50ndash67 2012

[31] X-S Si W Wang C-H Hu M-Y Chen and D-H Zhou ldquoAWiener-process-based degradationmodel with a recursive filteralgorithm for remaining useful life estimationrdquo MechanicalSystems and Signal Processing vol 35 no 1-2 pp 219ndash237 2013

[32] C-C Tsai S-T Tseng and N Balakrishnan ldquoMis-specificationanalyses of gamma andWiener degradation processesrdquo Journalof Statistical Planning and Inference vol 141 no 12 pp 3725ndash3735 2011

[33] F Haghighi and S J Bae ldquoReliability estimation from lineardegradation and failure time data with competing risks under astep-stress accelerated degradation testrdquo IEEE Transactions onReliability vol 64 no 3 pp 960ndash971 2015

[34] Z-S Ye Y Wang K-L Tsui and M Pecht ldquoDegradation dataanalysis usingwiener processeswithmeasurement errorsrdquo IEEETransactions on Reliability vol 62 no 4 pp 772ndash780 2013

[35] J Huang D S Golubovic S Koh et al ldquoDegradation modelingof mid-power white-light LEDs by using Wiener processrdquoOptics Express vol 23 no 15 pp A966ndashA978 2015

[36] A Beskos O Papaspiliopoulos G O Roberts and P Fearn-head ldquoExact and computationally efficient likelihood-basedestimation for discretely observed diffusion processes (withdiscussion)rdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 68 no 3 pp 333ndash382 2006

[37] J Shang and J E Cavanaugh ldquoAn assumption for the develop-ment of bootstrap variants of the Akaike information criterionin mixed modelsrdquo Statistics and Probability Letters vol 78 no12 pp 1422ndash1429 2008

[38] W QMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[39] W J Padgett andM A Tomlinson ldquoInference from accelerateddegradation and failure data based on Gaussian process mod-elsrdquo Lifetime Data Analysis vol 10 no 2 pp 191ndash206 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of