Research ArticleLikelihood Inference of Nonlinear Models Based on a Class ofFlexible Skewed Distributions

Xuedong Chen1 Qianying Zeng2 and Qiankun Song2

1 School of Science Huzhou Teachersrsquo College Huzhou 313000 China2 School of Management Chongqing Jiaotong University Chongqing 400074 China

Correspondence should be addressed to Qiankun Song qiankunsong163com

Received 28 August 2014 Accepted 28 September 2014 Published 3 December 2014

Academic Editor Jinde Cao

Copyright copy 2014 Xuedong Chen et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper deals with the issue of the likelihood inference for nonlinear models with a flexible skew-t-normal (FSTN) distributionwhich is proposed within a general framework of flexible skew-symmetric (FSS) distributions by combining with skew-t-normal(STN) distribution In comparison with the common skewed distributions such as skew normal (SN) and skew-t (ST) as well asscale mixtures of skew normal (SMSN) the FSTN distribution can accommodate more flexibility and robustness in the presenceof skewed heavy-tailed especially multimodal outcomes However for this distribution a usual approach of maximum likelihoodestimates based on EM algorithm becomes unavailable and an alternative way is to return to the original Newton-Raphson typemethod In order to improve the estimation as well as the way for confidence estimation and hypothesis test for the parametersof interest a modified Newton-Raphson iterative algorithm is presented in this paper based on profile likelihood for nonlinearregression models with FSTN distribution and then the confidence interval and hypothesis test are also developed Furthermorea real example and simulation are conducted to demonstrate the usefulness and the superiority of our approach

1 Introduction

The common assumption of distribution for random error isnormal in statistical modelingThis assumption may lack therobustness against departures from normality andor outliersand may result in misleading inferential results [1 2] For thepast few years there is an increasing interest in developingmore flexible parametric families capable of adopting asclosely as possible real data which exhibit quite substantialnonnormal characteristics such as skewness and heavy tailsIn a variety of applications one popular option is to modify asymmetric probability density function of a variable therebyintroducing skewness An important advantage of this sortof approach compared with other approaches to robustnessis an explicit statement of the probabilistic setting leading toa clear interpretation of the results [3] Following this ideathe skw-normal (SN) distribution was firstly introduced by[4] and then the skew-t (ST) distribution was introducedby [5] the skew-t-normal (STN) was introduced by [6]moreover some extensions to these multivariate cases werestudied by [7 8] and so on Since then several authors

have tried to extend these results to more general formsof skew-symmetric distributions of which here we wouldlike to mention [9] in this paper they proposed a generalframework of distributions which is called flexible skew-symmetric (FSS) distribution As pointed out by [10] that thisdistribution family enjoys a sufficient flexibility in that withdifferent choice of submodel settings the FSS distributionincludes several known distributions such as the SN and STas its special cases

However in many practical applications it is not rareat all to encounter a multimodality sometimes with aneven irregular shape and for this case all the distributionsmentioned above appear to be unsufficient to describe themultimodal feature of the data A solution to this problemis to use finite mixture models In [11] the authors workedwith a mixture model with component densities belongingto the STN distribution and a computationally feasible EM-type algorithm was developed for calculating the maximumlikelihood (ML) estimates of parameters Unfortunatelyalthough the proposed methodology is useful for analyzingmultimodal asymmetric data it suffers from the problem of

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 542985 8 pageshttpdxdoiorg1011552014542985

2 Abstract and Applied Analysis

ldquomodel identificationrdquo as the number of the parameters tobe estimated is usually large As a result in this paper wedeal with a new extension of the class of FSS distributionswhich is referred to as flexible skew-t-normal (FSTN) distri-bution This new distribution is proposed within the generalframework of the FSS distributions in combination with thedefinition of STN distribution In practical applications it isable to regulate the density in a more flexible way to offerrobustness and it can be treated as an appealing option foraccommodating data with skewness and heavy tails as well asmultimodality jointly

On the other hand nonlinear regression models arewidely applied in the fields of economics engineeringbiomedical research and so forth where the nonlinearfunction of unknown parameters is used to explain orinvestigate the nonlinear relationship of random phenomenaunder study More recently several authors have used aclass of skewed distributions in the context of nonlinearregression models and some valuable results were obtainedFor example [12] developed the robust estimation and thelocal influence analysis for regression model with SMSNdistribution From Bayesian point of view [13] consideredthe Bayesian estimation and the case influence diagnosticsfor nonlinear regression models with SMSN distributionsMore related literature could be found in [14ndash17] Generallyspeaking for model fitting of the nonlinear regression withskewed distributions a popular approach is to considerthe hierarchical representation of variables with a specificdistribution inwhich the postulated distribution is expressedas several conditional distributions of simpler forms suchas normal and Studentrsquos 119905 and Gamma Based on that EMalgorithm or Bayesian hierarchical approach then can beimplemented effectively for conducting model estimationand statistical inference

In this paper our aim is to develop an approach to like-lihood inference of nonlinear regression models with FSTNassumption As there is no stochastic representation for FSTNdistribution all the methods cited above become unavailablefor our considered problem and an alternativeway is to returnto the original Newton-Raphson iterative procedure formodel estimation Under the nonlinear regression paradigmthe accuracy of estimates is affected by the strength ofnonlinearity and the corresponding confidence interval andhypothesis test require the assumption of normality of theestimators or distribution which is too restrictive Besidesconsidering that in many practical applications rather thanthe total parameters we are usually interested in a propersubset of them By taking all these factors into accountin this paper we focus on the parameters of interest andpropose a modified Newton-Raphson iterative algorithm forcalculating the ML estimates based on profile likelihoodFurthermore the confidence interval and hypothesis test forthe parameters of interest are also considered We conductan application and a simulation study to compare the algo-rithm effectiveness and distribution robustness for nonlinearregression model in terms of fitting performance and modelselection The results from the numerical examples illustratethe usefulness and the superiority of our methodology

The remainder of this paper is organized as followsIn Section 2 we briefly discuss the FSS distribution andFSTN distributions In Section 3 we present the likelihoodinference including the quantities of the first- and the second-order derivatives as well as the standard Newton-Raphsoniterative formula In Section 4 we give an introductionof profile inference for our proposed model where theconfidence estimation and hypothesis test are presented tooSection 5 gives numerical examples using both simulatedand real data to illustrate the performance of the proposedmethodology Finally some concluding remarks are given inSection 6

2 Models and Notation

The class of skewed distributions such as SN ST and STNperform to be plausible for modeling skewness or (and)heavy tails underlying the observations The actual situationis that it is not rare at all to encounter multimodalitysometimes with an even more irregular shape and for thiscase the aforementioned distributions become unsufficientIn this paper with the adoption of a sufficiently flexibleclass of distributions we consider one of these extensionsreferred to as the family of flexible skew-symmetric (FSS)distributions which is introduced by [9] with the followingdensity function of type

119891 (119910 120583 120596120572) = 2120596minus1

1198910

(119911) 119866 119875119870

(119911) (1)

where 1198910

and 119866 are symmetric univariate density and dis-tribution function respectively that is 119891

0

(119909) = 1198910

(minus119909)119866(minus119909) = 1minus119866(119909) and 119875

119870

(119909) = 1205721

119909+1205723

1199093

+ sdot sdot sdot 1205722119870minus1

1199092119870minus1 is

an odd polynomial of degree 119870 (ie a polynomial includingonly terms of odd degree) 119911 = 120596

minus1

(119909 minus 120583) 120583 isin 119877 is thelocation parameter 120596 gt 0 is the scale parameter and 120572 =

(1205721

1205723

1205722119870minus1

) and1205721

1205723

1205722119870minus1

isin 119877 are shape param-eters

In general the density function of STN distribution canbe represented as 2120596

minus1

119905(119911 V)Φ(120572119911) where 119905 and Φ respec-tively denote the univariate standard Studentrsquos 119905 densityfunction and the univariate standard normal distributionfunction and V is the degrees of freedom The skewness isregulated by the shape parameter 120572 and the tail thickness ofthe distribution is controlled by V Commonly in comparisonwith the SN distribution the STN distribution exhibitsobvious feature of heavy tails when V le 10

In this paper we work with one version of (1) and thespecific definition can be presented as follows Let 119891

0

(119911) =

119905(119911 V) and 119866(119909) = Φ(119909) where Φ(119909) is defined as before and

119905 (119911 V) =Γ (V + 1) 2

120596 (V120587)12

Γ (V2)

(1 +1199112

V)

minus((V+1)119889)

(2)

that is the density function of univariate Studentrsquos 119905 distribu-tion with 0 location 120596 scale and V degrees of freedom Theabove extension of (1) is referred to as flexible skew-t-normal(FSTN) distribution denoted by FSTN(120583 120596120572 V)

It is noted that the FSTN distribution is proposed withinthe general framework of FSS distribution by combining with

Abstract and Applied Analysis 3

the definition of STN distribution and as a consequence itshares analogous feature with these two distributions Forall of that the FSTN distribution presents some interestingand peculiar features and is able to regulate the density ina more flexible way To be particular except for modelingskewness and tail thickness the FSTN distribution allows formultimodality depending on the specific setting of 120572 For thepurpose of comparison and illustration we assume 119891

0

(119911) =

120601(119911) and 119866(119911) = Φ(119911) in FSS distribution which is denotedby FSN and then we set 119870 = 2 in 119875

119870

(119911) that is 119875119870

(119909) =

1205721

119909 + 1205723

1199093 in STN distribution and for this case different

selections of 1205721

and 1205723

determine whether the density isunimodal or bimodal Moreover the same assumption for 119866

and 119875 in FSTN distribution is madeFigure 1 displays the density functions of FSN and STN

as well as FSTN distributions with four different situationsconsidered namely 120572

1

= 1 1205723

= 0 and V = 10 1205721

= 1205723

= 1

and V = 6 1205721

= 1 1205723

= minus1 and V = 4 1205721

= minus11205723

= 1 and V = 4 respectively with 120583 = 1 120596 = 15 for allcases By examination of Figure 1 we can detect how thesethree densities change with different combinations of 120572 andV For instance in Figure 1(a) FSN STN and FSTN appearto be very close while STN and FSTN are heavy tailed to alittle extent In Figure 1(b) both FSN and FSTN are unimodalwhen 120572

1

and 1205723

keep the same sign and the ranking for thedegree of skewness is STN FSTN and FSN in turn that isthe STN and FSTN distributions have thicker tails comparedto FSN distribution With opposite sign of 120572

1

and 1205723

bothFSN and FSTN distributions are bimodal and highly skewedin Figures 1(c) and 1(d) moreover in the same directionFSTN has thicker tails than FSN distribution Our proposedFSTN distribution can be treated as a proper compromisebetween the FSS distribution and the STN distribution Itallows for a wider range of tail behavior compared to FSSdistributionwhereas it is able to accommodatemultimodalitywhich cannot be described by STN distribution From theapplied viewpoint the FSTN distribution is an appealingoption which can be expected to yield robust inferentialresults in the presence of outlying observations

3 Likelihood Inference

Consider 119899 independent observations satisfying a nonlinearregression model as [119910

119894

] sim FSTN(120583119894

120596120572 V) with 120583119894

= 120578(x119894

120573) for 119894 = 1 2 119899 Here x

119894

is a 119901-dimensional vector and 120573is a 119901 times 1 vector of parameters Also let X = (x

1

x119899

)119879 be

the 119899 times 119901 design matrix 120578(sdot sdot) is a known twice differentiablefunction Then the corresponding log-likelihood for param-eter 120579 = (120573120572 120596 V) is given by

119897 (120579) = constant minus 119899 log120596 +

119899

sum

119894=1

log 119905 (119911119894

) +

119899

sum

119894=1

logΦ 119875119870

(119911119894

)

(3)And we have

120597119897 (120579)

120597120573= minus

1

120596

119899

sum

119894=1

(1199051015840

(119911119894

)

119905 (119911119894

)+ 1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

))

120597120578119894

120597120573

120597119897 (120579)

120597120596= minus

1

120596

119899

sum

119894=1

(1199051015840

(119911119894

)

119905 (119911119894

)119911119894

+ 119911119894

1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)+ 119899)

120597119897 (120579)

120597V=

119899

sum

119894=1

1199112

119894

119905 (119911119894

)+

119899

2(119863119866 (

V + 1

2) minus 119863119866 (

V2

) minus1

2)

120597119897 (120579)

1205971205722119895minus1

=

119899

sum

119894=1

1199112119895minus1

119894

120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)1198751015840

119870

(119911119894

) 119895 = 1 2 119870

(4)

where 120578119894

= 120578(x119894

120573) and 119863119866(119909) = (119889[log Γ(119909)])119889119909 The cor-responding second-order derivatives of (3) can be shown as

1205972

119897 (120579)

120597120573120597120573119879

=

119899

sum

119894=1

(120597120578119894

120597120573)

119879

1

1205962

[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

)

+ 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

120597120578119894

120597120573

minus1

120596[

1199051015840

(119911119894

)

119905 (119911119894

)+ 1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

1205972

120578119894

120597120573120597120573119879

1205972

119897 (120579)

120597120573120597120596=

119899

sum

119894=1

1

1205962

[1199051015840

(119911119894

)

119905 (119911119894

)+ 1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

+119911119894

1205962

[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

)

+ 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

120597120578119894

120597120573

1205972

119897 (120579)

120597120573120597V= minus

1

120596

119899

sum

119894=1

[

(1 + Vminus1) 1199113

119894

minus 119905 (119911119894

) 1199112

119894

V21199052 (119911119894

)]

120597120578119894

120597120573

1205972

119897 (120579)

1205971205731205971205722119895minus1

=

119899

sum

119894=1

1199112119895minus1

119894

[1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

) + 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

times120597120578119894

120597120573 119895 = 1 2 119870

1205972

119897 (120579)

1205971205962

=1

1205962

119899

sum

119894=1

[1199051015840

(119911119894

)

119905 (119911119894

)+ 1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)] 3119911119894

+ 1199112

119894

[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

)

+ 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

1205972

119897 (120579)

120597120596120597V= minus

1

120596

119899

sum

119894=1

[

(1 + Vminus1) 1199113

119894

minus 119905 (119911119894

) 1199112

119894

V21199052 (119911119894

)]

1205972

119897 (120579)

1205971205961205971205722119895minus1

= minus

119899

sum

119894=1

[1

120596(1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

) + 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)) 119911119894

]

times 1199112119895minus1

119894

119895 = 1 2 119870

4 Abstract and Applied Analysis

000

005

010

015

020

025

030

x

Density

FSNSTNFSTN

minus2 0 2 4 6

(a)

x

FSNSTNFSTN

000

005

010

015

020

025

030

035

Density

minus2 0 2 4 6

(b)

x

FSNSTNFSTN

minus2 0

000

005

010

015

020

025

030

Density

2 4 6

(c)

x

FSNSTNFSTN

000

005

010

015

020

025

030

Density

minus2 0 2 4 6

(d)

Figure 1 ((a)ndash(d)) Density functions of FSN STN and FSTN distributions with four different situations

1205972

119897 (120579)

120597V2=

119899

2[119879119866 (

V + 1

2) minus 119879119866 (

V2

)] +

119899

sum

119894=1

log (119905 (119911119894

))

1205972

119897 (120579)

1205971205722119895minus1

1205971205722119896minus1

=

119899

sum

119894=1

119911119895+119896

119894

119876 119875119870

(119911119894

) 119895 119896 = 1 2 119870

1205972

119897 (120579)

120597V1205971205722119895minus1

= 0 119895 = 1 2 119870

(5)

where 119879119866(119909) = (1198892

[log Γ(119909)])1198891199092 ℎ(119909) = (119905

10158401015840

(119909)119905(119909) minus

[1199051015840

(119909)]2

)1199052

(119909) and 119876(119909) = (120601(119909)Φ(119909))[119909 + (120601(119909)Φ(119909))]Assume119880(120579) = (119880

119879

(120573) 119880(120596) 119880(V) 119880119879(120572))119879 is the gradi-ent or score vector and 119867(120579) is the Hessian matrix composedof the above second-order derivatives To obtain the MLestimate of 120579 the Newton-Raphson iteration algorithm isdefined by

(119896+1)

= (119896)

minus [119867(120579)]

minus1

120579=

120579

(119896) [119880(120579)]120579=

120579

(119896) (6)

Abstract and Applied Analysis 5

It is noted that the above iterative procedure is an unpar-titioned algorithm that is all the parameters includingnonlinear regression coefficients 120573 scale parameter 120596 andshape parameter 120572 as well as tail thickness parameter V areestimated simultaneously For our considered problem thereare at least two difficulties that may be encountered for (6)the first one is that once the number of the parameters to beestimated becomes large the corresponding computationalburden turns to be heavy with an unacceptable estimationerror and the second one is as follows when the strength ofnonlinearity of the link function 120578(sdot sdot) changes the iterativeprocessmay becomeunstable or even nonconvergent leadingto the poor estimation results Considering the needs ofpractical problems rather than the total parameter set weare usually interested in a proper subset of it To improvethe efficiency of the algorithm and to facilitate statisticalinference of the nonlinearmodels with FSTNdistribution weput forward the following profile likelihoodmethod based on(3) and (6)

4 Profile Likelihood Inference

41 Profile Estimation Algorithm Let 120579 = (120573119879

120574119879

) be a parti-tion of 120579 where 120573 is a parameter vector of nonlinear regres-sion an interest parameter and 120574 is a 119902-dimension (119902 = 119870+

2) nuisance parameter with 120574 = (120572 120596 V)119879 Similarly thepartition of 119867 and 119880 is given as 119867(120579) = (

11986711 11986712

11986721 11986722) where

11986711

= (1205972

119897(120579)120597120573120597120573119879

) 11986712

= (1205972

119897(120579)120597120573120597120574119879

) = (1205972

119897(120579)

120597120573120597120596 1205972

119897(120579)120597120573120597V 1205972119897(120579)1205971205731205971205721

1205972

119897(120579)1205971205731205971205722119896minus1

) 11986712

= [11986721

]119879 and 119867

22

= (1205972

119897(120579) 120597120574120597120574119879

) and the diagonalelements of 119867

22

are given by 1205972

119897(120579) 1205971205962

1205972

119897(120579)120597V21205972

119897(120579)1205971205722

1

1205972

119897(120579)1205971205722

2119896minus1

respectively and the off-diagonal elements in 119867

22

can be obtained similarly Let119880(120579) = (119880

1

1198802

) corresponding to the partition of 120579In the subsequent context we focus on the estimation of120579 based on profile likelihood method [18] Firstly suppose 120573is known and we rewrite the original likelihood function (3)as

119897 (120579) = 119897 (120573 120574) = 119897120573

(120574) (7)

where notation 119897120573

(120574) denotes that 120573 is fixed but 120574 varies Foreach 120573 to estimate 120574 we can obtain

120573

= argmax120574

119897120573

(120574) (8)

Alternatively to estimate 120573 we evaluate the maximum valueof 119897120573

(120574) over 120573

and have

119901

= argmax120573

119897120573

(120573

) = argmax120573

119897 (120573 120573

) (9)

where 119897(120573 120573

) and 119901

are referred to as the profile likelihoodfunction and the profile ML estimation respectively

Following [19] we define the profile Newton-Raphsoniteration formula as follows

(119896+1)

119901

= (119896)

119901

minus [119867120573|120574

]minus1

(119880120573|120574

)

(119896+1)

119901

= (119896)

119901

minus ((11986722

)minus1

1198801

+ (11986722

)minus1

11986721

[119867120573|120574

]minus1

(119880120573|120574

))

(10)

where 119880120573|120574

= 1198801

minus 11986712

(11986722

)minus1

1198802

119867120573|120574

= 11986711

minus

11986712

(11986722

)minus1

11986721

and all the matrices and the vectors on theright hand side of (10) are evaluated at

(119896)

119901

and (119896)119901

Note that both in (6) and in (10) the strength of

nonlinearity of link function 120578(sdot sdot) is reflected by 1205972

120578119894

120597120573120597120573119879

to large extents Therefore by examination of the expressionof 1205972

119897(120579)120597120573120597120573119879 we find that when the element of

100381610038161003816100381610038161003816100381610038161003816

120596 [1199051015840

(119911119894

)

119905 (119911119894

)+ 1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

1205972

120578119894

120597120573120597120573119879

100381610038161003816100381610038161003816100381610038161003816

(11)

is much less than the corresponding element of1003816100381610038161003816100381610038161003816100381610038161003816

120597120578119879

119894

120597120573[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

) + 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

times120597120578119894

120597120573

100381610038161003816100381610038161003816100381610038161003816

(12)

the following approximating result can be obtained as

119867lowast

11

=

119899

sum

119894=1

(120597120578119894

120597120573)

119879

1

1205962

[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

)

+ 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

120597120578119894

120597120573

(13)

And then the iteration formulas of 119880lowast

120573|120574

[119867lowast

120573|120574

]minus1 119901

and

lowast

119901

can also be obtained just as before The above estimationprocedure is referred to as profile modified Newton-Raphsoniteration algorithm

The stopping rule for the above algorithm can bepresented as that iteration proceeds until some distanceinvolving two successive evaluations of the profile likeli-hood 119897(120573

120573

) such as |119897((119896+1)

120573

(119896+1)) minus 119897((119896)

120573

(119896))| or

|119897((119896+1)

120573

(119896+1))119897((119896)

120573

(119896))minus1| is small enough for example120576 lt 10

minus4 is adopted in this paperThe asymptotic covariancematrix of theML estimates for

profile likelihood can be evaluated by inverting the expectedinformation matrix however it does not have a closedform expression the observed information matrix 119869(120573) =

minus(1205972

119897(120573 120573

)120597120573120597120573119879

) can be used as a replacement which isestimated by 119867

11

minus 11986712

(11986722

)minus1

11986721

where 11986711

11986712

and 11986722

can be obtained as similar as aboveThe choice of the initial values plays an important role in

nonlinear regression fitting in this paper the specific steps

6 Abstract and Applied Analysis

for choosing the starting values are implemented as follows

(i) compute the initial value (0)

based on the nonlinearregression model with standard normal assumption

(ii) with (0)

fixed compute the initial values 120596(0) and

120572(0) for the SN and finite mixture SN assumptions

respectively

In order to simplify the estimation of parameter V wehave fixed integral values for V from 2 to 40 by one choose thevalue of V that maximizes the profile likelihood as V(0) andthen the initial values of

(0)

120596(0) 120572(0) V(0) that are requiredin the estimation procedure are all obtained

42 Profile Confidence Estimation and Hypothesis Test Con-fidence interval and hypothesis test play an important role instatistical inference and in the subsequent content we willconsider the profile confidence estimation andhypothesis testfor the parameters of interest in nonlinear regressionmodelsSuppose 119869(120579) = 119864[minus119867(120579)] the following regular conditionsfor likelihood inference are assumed

(R-i) int(120597 log 119871(x 120579)120597120579) sdot 119871(x 120579)119889x = 0

(R-ii) (120597120597120579) int 119871(x 120579) = int(120597119871(x 120579)120597120579)119889x = 0

(R-iii) (120597120597120579) int 119892(x)119871(x 120579)119889x = int 119892(x)(120597119871(x 120579)120597120579)119889xwhere 119892(x) is arbitrary measurable function

(R-iv) 119864(120597 log 119871(x 120579)120597120579)2

gt 0

Apart from the above assumptions in this paper someadditional conditions are assumed to hold for that

(A-i) 120578(x119894

120573) is twice continuously differentiable withrespect to 120573 isin 119877

119901(A-ii) 119863 = (120597120578

1

120597120573 120597120578119899

120597120573)119879 is full rank for all 120573

(A-iii) assume 1205790

is the true value of 120579 then there exist aneighbour region 119873

0

of 1205790

and a constant 119888 such that120582min(119869(120579)) gt 119888 for any 120579 isin 119873

0

Under the above assumption following [20] we have

2 119897 (119901

120573119901) minus 119897 (120573

0

1205730

) 997888rarr 1205942

(119901) (14)

where 1205730

is the true value of 120573 and the convergence is underthe meaning of convergence in probability Based on theprofile likelihood theory the confidence region of 120573 with theconfidence level 100(1 minus 120572) is given by

119862 (119884) = 120573 2 [119897 (119901

120573119901) minus 119897 (120573

120573

)] le 1205942

(119901 120572) (15)

where 1205942

(119901 120572) denotes the upper 120572 percentile of chi-squaredistribution with 119901 degrees of freedom Furthermore thehypothesis test

1198670

120573 = 1205730

against 1198671

120573 = 1205730

(16)

can also be considered and the corresponding test statisticsare presented by 2[119897(

120573

) minus 119897(1205730

1205730

)] Unlike standard

likelihood method profile likelihood confidence intervalsand hypothesis test do not need assumption of normality ofthe estimator or distribution which is too restrictive they arebased on an asymptotic chi-square distribution of the log pro-file likelihood ratio test statistics and these properties bring alot of convenience and feasibility in practical computation

5 Simulation Study

To investigate the experimental performance of our method-ology we undertake a simulation study to compare the fittingperformance of misspecified distribution as well as under-standing the large sample properties of the ML estimates Torealize this purpose we generate the artificial data from thefollowing two nonlinear models

Model (1) drug responsiveness model

119910119894

= 1205730

minus1205730

1 + (119909119894

1205731

)1205732

+ 120576119894

(17)

Model (2) nonlinear growth-curve model

119910119894

=1205731

1 + 1205732

exp (minus1205733

119909)+ 120576119894

(18)

with two kinds of skewed distributions for random error asfollows Case (I) 120576 sim STN(120583 120596 120572 V) and Case (II) 120576 sim

FSTN(120583 120596120572 V)The true parameter population is chosen as follows 120573 =

(2 5 minus08) is assumed to be the same for the above twomodels In Model (1) let 120583 = 120575

1

120596 = 15 120572 = 15 and V = 4

and in Model (2) let 120583 = 1205752

120596 = 15 120572 = (1 minus08) and V = 4Note that for the values of the degrees of freedom V a relativevalue (V = 4) can yield a heavy-tailed distribution as we need

In this simulation there are totally four simulated datasets corresponding to two nonlinear models Model (1) andModel (2) along with two distributions Case (I) and Case(II) for random error Similar to previous analysis eachsimulated data set is fitted under STN and FSN as well asFSTN scenarios using three different estimation algorithmsTo shed light on the experimental performance of ourmethodology an interesting comparison can be made byexamining how often we can recognize the true model

Table 1 shows the absolute value of the average biasbetween the true and the estimated parameters and thepercentages of each model being ranked as the best modelbased on AIC criterion out of 500 replications are alsopresented in Table 1

By examination of Table 1 we can find the following(i) the difference for the estimation results between drugresponsiveness model and growth-curve model is significantindicating that the nonlinearity of model imposes an impacton parameter estimation (ii) when the true distribution forrandom error is STN the three fitting distributions haveroughly the same behavior it is hard to tell which approachfor parameter estimation is good and which approach isbad and the similar results can be obtained for modelselection based on AIC (iii) when the true distribution forrandom error is FSTN the FSN and the FSTN distributions

Abstract and Applied Analysis 7

Table 1 Results of the simulation study

Method Model Error Fittingmodel

Bias and perc1205730

1205731

1205732

()

NR

1

Case ISTN 0048 0044 0034 384

FSN 0052 0043 0031 302

FSTN 0051 0048 0035 314

Case IISTN 0941 0445 0374 105

FSN 0035 0014 0011 438

FSTN 0033 0010 0010 457

2

Case ISTN 0067 0061 0053 362

FSN 0070 0057 0055 302

FSTN 0073 0072 0048 336

Case IISTN 0963 0469 0391 85

FSN 0047 0023 0017 446

FSTN 0043 0022 0021 469

P-NR

1

Case ISTN 0049 0040 0032 326

FSN 0050 0047 0038 401

FSTN 0052 0049 0036 273

Case IISTN 1029 0420 0248 114

FSN 0039 0018 0010 396

FSTN 0038 0017 0009 490

2

Case ISTN 0070 0050 0047 303

FSN 0067 0057 0056 423

FSTN 0073 0071 0050 274

Case IISTN 1052 0446 0265 102

FSN 0053 0031 0020 396

FSTN 0052 0022 0017 502

MP-NR

1

Case ISTN 0045 0049 0033 323FSN 0051 0047 0036 383FSTN 0045 0044 0031 294

Case IISTN 0773 0373 0266 145FSN 0031 0018 0011 384FSTN 0028 0010 0008 471

2

Case ISTN 0068 0072 0062 281FSN 0076 0074 0062 412FSTN 0073 0060 0054 307

Case IISTN 0794 0394 0290 113FSN 0037 0026 0017 394FSTN 0036 0021 0013 493

outperform the STNdistribution by producing estimateswithlower bias and higher AIC proportion furthermore the PNRand MPNR methods perform better than the traditional NRmethod in general

To study the consistence properties of ML estimate wefocus on the situation that the true distribution for randomerror is Case (II) whereas the fitting distribution is FSTN tooFor this case two estimation algorithms PNR and MPNR

are adopted for parameter estimation and samples of differentsizes (119899 = 50 100 and 200) are generated from Models (1)

and (2) We compute the 95 confidence interval for theparameters of interest and the mean square error (MSE) fordifferent model where MSE(120573) = (1500) sum

500

119894=1

120573(119894)

minus 1205732

The length of the 95 confidence interval and theMSE resultsare summarized in Table 2 From Table 2 we can see thatthe length of the confidence interval and the MSE tend todecrease as the sample size increases as expected

Tables 1 and 2 show that in general FSTN distributionenjoys more robustness and flexibility in modeling data withskewness and heavy tails as well asmultimodality in compari-sonwith other skewed alternatives and the implementation ofMPNR method brings more accuracy and improvement formodel estimation in the context of nonlinear regression withthis new distribution

6 Conclusion

We have proposed a new skewed distribution based on thegeneral FSS distribution framework called the FSTN dis-tribution which is allowed to accommodate multimodalityasymmetry and heavy tails jointly to offer greater flexibilitythan SN and STN counterpartsMoreover we have developeda modified profile of Newton-Raphson iterative algorithmfor estimating the parameters of interest of nonlinear modelwith FSTN distribution and the interval estimation andhypothesis test in a profile likelihood paradigm are alsoconsidered

Numerical studies reveal that in the context of nonlinearregression analysis if the true distribution is STN or FSNwhereas the fitting distribution is FSTN the estimationresults are not influenced by this misspecification of distribu-tion assumptionHowever once the true distribution is FSTNwhile the fitting distribution is STN the estimation resultsappear to be somewhat disappointing which shows therobustness of FSTN distribution In general the combinationof FSTN distribution with MPNR method brings moreaccuracy and improvement on the estimation of nonlinearregression

So far the present methodology is limited to the completedata analysis the extensions of this paper include missingdata version as well as Bayesian analysis of this model whichwill be reported in another paper

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 11171105 and 61273021 andin part by the Natural Science Foundation Project of CQcstc2013jjB40008

8 Abstract and Applied Analysis

Table 2 The table shows the length of 95 confidence interval of parameters of interest and MSE of specified model

Model Sample size Profile-NR Modified profile-NR1205730

1205731

1205732

MSE 1205730

1205731

1205732

MSE

150 1015 0675 0374 0195 0959 0589 0341 0167100 0558 0365 0152 0051 0519 0254 0132 0073200 0172 0134 0076 0004 0120 0141 0081 0005

250 1258 0828 0449 0240 1152 0692 0390 0154100 0569 0448 0208 0099 0516 0427 0211 0022200 0232 0181 0122 0009 0214 0145 0120 0008

References

[1] G Verbeke and E Lesaffre ldquoA linear mixed-effects model withheterogeneity in the random-effects populationrdquo Journal of theAmerican Statistical Association vol 91 no 433 pp 217ndash2211996

[2] P GhoshMD Branco andH Chakraborty ldquoBivariate randomeffectmodel using skew-normal distributionwith application toHIV-RNArdquo Statistics in Medicine vol 26 no 6 pp 1255ndash12672007

[3] A Azzalini and M G Genton ldquoRobust likelihood methodsbased on the skew-t and related distributionsrdquo InternationalStatistical Review vol 76 no 1 pp 106ndash129 2008

[4] A Azzalini ldquoA class of distributions which includes the normalonesrdquo Scandinavian Journal of Statistics vol 12 no 2 pp 171ndash178 1985

[5] A Azzalini and A Capitanio ldquoDistributions generated byperturbation of symmetrywith emphasis on amultivariate skew119905-distributionrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 65 no 2 pp 367ndash389 2003

[6] H W Gomez O Venegas and H Bolfarine ldquoSkew-symmetricdistributions generated by the distribution function of thenormal distributionrdquo Environmetrics vol 18 no 4 pp 395ndash4072007

[7] A Azzalini and A Dalla Valle ldquoThe multivariate skew-normaldistributionrdquo Biometrika vol 83 no 4 pp 715ndash726 1996

[8] A Azzalini and A Capitanio ldquoStatistical applications of themultivariate skew normal distributionrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 61 no 3 pp579ndash602 1999

[9] Y Ma and M G Genton ldquoFlexible class of skew-symmetricdistributionsrdquo Scandinavian Journal of Statistics vol 31 no 3pp 459ndash468 2004

[10] A Azzalini ldquoThe skew-normal distribution and related multi-variate familiesrdquo Scandinavian Journal of Statistics vol 32 no2 pp 159ndash200 2005

[11] H J Ho S Pyne and T I Lin ldquoMaximum likelihood inferencefor mixtures of skew Student-119905-normal distributions throughpractical EM-type algorithmsrdquo Statistics andComputing vol 22no 1 pp 287ndash299 2012

[12] C B Zeller VH Lachos and F E Vilca-Labra ldquoLocal influenceanalysis for regression models with scale mixtures of skew-normal distributionsrdquo Journal of Applied Statistics vol 38 no2 pp 343ndash368 2011

[13] V G Cancho D K Dey V H Lachos and M G AndradeldquoBayesian nonlinear regression models with scale mixturesof skew-normal distributions estimation and case influencediagnosticsrdquo Computational Statistics amp Data Analysis vol 55no 1 pp 588ndash602 2011

[14] M D Branco and D K Dey ldquoA general class of multivariateskew-elliptical distributionsrdquo Journal of Multivariate Analysisvol 79 no 1 pp 99ndash113 2001

[15] F C Xie J G Lin and B CWei ldquoDiagnostics for skew-normalnonlinear regression models with AR(1) errorsrdquo ComputationalStatistics and Data Analysis vol 53 no 12 pp 4403ndash4416 2009

[16] F-C Xie B-C Wei and J-G Lin ldquoHomogeneity diagnosticsfor skew-normal nonlinear regressions modelsrdquo Statistics ampProbability Letters vol 79 no 6 pp 821ndash827 2009

[17] L H Vanegas and F J Cysneiros ldquoAssessment of diagnosticprocedures in symmetrical nonlinear regression modelsrdquo Com-putational Statistics amp Data Analysis vol 54 no 4 pp 1002ndash1016 2010

[18] T A Severini ldquoAn approximation to the modified profilelikelihood functionrdquo Biometrika vol 85 no 2 pp 403ndash4111998

[19] G K Smyth ldquoPartitioned algorithms for maximum likelihoodand other non-linear estimationrdquo Statistics and Computing vol6 no 3 pp 201ndash216 1996

[20] B Wei Exponential Family Nonlinear Models Springer Singa-pore 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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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

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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

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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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

2 Abstract and Applied Analysis

ldquomodel identificationrdquo as the number of the parameters tobe estimated is usually large As a result in this paper wedeal with a new extension of the class of FSS distributionswhich is referred to as flexible skew-t-normal (FSTN) distri-bution This new distribution is proposed within the generalframework of the FSS distributions in combination with thedefinition of STN distribution In practical applications it isable to regulate the density in a more flexible way to offerrobustness and it can be treated as an appealing option foraccommodating data with skewness and heavy tails as well asmultimodality jointly

On the other hand nonlinear regression models arewidely applied in the fields of economics engineeringbiomedical research and so forth where the nonlinearfunction of unknown parameters is used to explain orinvestigate the nonlinear relationship of random phenomenaunder study More recently several authors have used aclass of skewed distributions in the context of nonlinearregression models and some valuable results were obtainedFor example [12] developed the robust estimation and thelocal influence analysis for regression model with SMSNdistribution From Bayesian point of view [13] consideredthe Bayesian estimation and the case influence diagnosticsfor nonlinear regression models with SMSN distributionsMore related literature could be found in [14ndash17] Generallyspeaking for model fitting of the nonlinear regression withskewed distributions a popular approach is to considerthe hierarchical representation of variables with a specificdistribution inwhich the postulated distribution is expressedas several conditional distributions of simpler forms suchas normal and Studentrsquos 119905 and Gamma Based on that EMalgorithm or Bayesian hierarchical approach then can beimplemented effectively for conducting model estimationand statistical inference

In this paper our aim is to develop an approach to like-lihood inference of nonlinear regression models with FSTNassumption As there is no stochastic representation for FSTNdistribution all the methods cited above become unavailablefor our considered problem and an alternativeway is to returnto the original Newton-Raphson iterative procedure formodel estimation Under the nonlinear regression paradigmthe accuracy of estimates is affected by the strength ofnonlinearity and the corresponding confidence interval andhypothesis test require the assumption of normality of theestimators or distribution which is too restrictive Besidesconsidering that in many practical applications rather thanthe total parameters we are usually interested in a propersubset of them By taking all these factors into accountin this paper we focus on the parameters of interest andpropose a modified Newton-Raphson iterative algorithm forcalculating the ML estimates based on profile likelihoodFurthermore the confidence interval and hypothesis test forthe parameters of interest are also considered We conductan application and a simulation study to compare the algo-rithm effectiveness and distribution robustness for nonlinearregression model in terms of fitting performance and modelselection The results from the numerical examples illustratethe usefulness and the superiority of our methodology

The remainder of this paper is organized as followsIn Section 2 we briefly discuss the FSS distribution andFSTN distributions In Section 3 we present the likelihoodinference including the quantities of the first- and the second-order derivatives as well as the standard Newton-Raphsoniterative formula In Section 4 we give an introductionof profile inference for our proposed model where theconfidence estimation and hypothesis test are presented tooSection 5 gives numerical examples using both simulatedand real data to illustrate the performance of the proposedmethodology Finally some concluding remarks are given inSection 6

2 Models and Notation

The class of skewed distributions such as SN ST and STNperform to be plausible for modeling skewness or (and)heavy tails underlying the observations The actual situationis that it is not rare at all to encounter multimodalitysometimes with an even more irregular shape and for thiscase the aforementioned distributions become unsufficientIn this paper with the adoption of a sufficiently flexibleclass of distributions we consider one of these extensionsreferred to as the family of flexible skew-symmetric (FSS)distributions which is introduced by [9] with the followingdensity function of type

119891 (119910 120583 120596120572) = 2120596minus1

1198910

(119911) 119866 119875119870

(119911) (1)

where 1198910

and 119866 are symmetric univariate density and dis-tribution function respectively that is 119891

0

(119909) = 1198910

(minus119909)119866(minus119909) = 1minus119866(119909) and 119875

119870

(119909) = 1205721

119909+1205723

1199093

+ sdot sdot sdot 1205722119870minus1

1199092119870minus1 is

an odd polynomial of degree 119870 (ie a polynomial includingonly terms of odd degree) 119911 = 120596

minus1

(119909 minus 120583) 120583 isin 119877 is thelocation parameter 120596 gt 0 is the scale parameter and 120572 =

(1205721

1205723

1205722119870minus1

) and1205721

1205723

1205722119870minus1

isin 119877 are shape param-eters

In general the density function of STN distribution canbe represented as 2120596

minus1

119905(119911 V)Φ(120572119911) where 119905 and Φ respec-tively denote the univariate standard Studentrsquos 119905 densityfunction and the univariate standard normal distributionfunction and V is the degrees of freedom The skewness isregulated by the shape parameter 120572 and the tail thickness ofthe distribution is controlled by V Commonly in comparisonwith the SN distribution the STN distribution exhibitsobvious feature of heavy tails when V le 10

In this paper we work with one version of (1) and thespecific definition can be presented as follows Let 119891

0

(119911) =

119905(119911 V) and 119866(119909) = Φ(119909) where Φ(119909) is defined as before and

119905 (119911 V) =Γ (V + 1) 2

120596 (V120587)12

Γ (V2)

(1 +1199112

V)

minus((V+1)119889)

(2)

that is the density function of univariate Studentrsquos 119905 distribu-tion with 0 location 120596 scale and V degrees of freedom Theabove extension of (1) is referred to as flexible skew-t-normal(FSTN) distribution denoted by FSTN(120583 120596120572 V)

It is noted that the FSTN distribution is proposed withinthe general framework of FSS distribution by combining with

Abstract and Applied Analysis 3

the definition of STN distribution and as a consequence itshares analogous feature with these two distributions Forall of that the FSTN distribution presents some interestingand peculiar features and is able to regulate the density ina more flexible way To be particular except for modelingskewness and tail thickness the FSTN distribution allows formultimodality depending on the specific setting of 120572 For thepurpose of comparison and illustration we assume 119891

0

(119911) =

120601(119911) and 119866(119911) = Φ(119911) in FSS distribution which is denotedby FSN and then we set 119870 = 2 in 119875

119870

(119911) that is 119875119870

(119909) =

1205721

119909 + 1205723

1199093 in STN distribution and for this case different

selections of 1205721

and 1205723

determine whether the density isunimodal or bimodal Moreover the same assumption for 119866

and 119875 in FSTN distribution is madeFigure 1 displays the density functions of FSN and STN

as well as FSTN distributions with four different situationsconsidered namely 120572

1

= 1 1205723

= 0 and V = 10 1205721

= 1205723

= 1

and V = 6 1205721

= 1 1205723

= minus1 and V = 4 1205721

= minus11205723

= 1 and V = 4 respectively with 120583 = 1 120596 = 15 for allcases By examination of Figure 1 we can detect how thesethree densities change with different combinations of 120572 andV For instance in Figure 1(a) FSN STN and FSTN appearto be very close while STN and FSTN are heavy tailed to alittle extent In Figure 1(b) both FSN and FSTN are unimodalwhen 120572

1

and 1205723

keep the same sign and the ranking for thedegree of skewness is STN FSTN and FSN in turn that isthe STN and FSTN distributions have thicker tails comparedto FSN distribution With opposite sign of 120572

1

and 1205723

bothFSN and FSTN distributions are bimodal and highly skewedin Figures 1(c) and 1(d) moreover in the same directionFSTN has thicker tails than FSN distribution Our proposedFSTN distribution can be treated as a proper compromisebetween the FSS distribution and the STN distribution Itallows for a wider range of tail behavior compared to FSSdistributionwhereas it is able to accommodatemultimodalitywhich cannot be described by STN distribution From theapplied viewpoint the FSTN distribution is an appealingoption which can be expected to yield robust inferentialresults in the presence of outlying observations

3 Likelihood Inference

Consider 119899 independent observations satisfying a nonlinearregression model as [119910

119894

] sim FSTN(120583119894

120596120572 V) with 120583119894

= 120578(x119894

120573) for 119894 = 1 2 119899 Here x

119894

is a 119901-dimensional vector and 120573is a 119901 times 1 vector of parameters Also let X = (x

1

x119899

)119879 be

the 119899 times 119901 design matrix 120578(sdot sdot) is a known twice differentiablefunction Then the corresponding log-likelihood for param-eter 120579 = (120573120572 120596 V) is given by

119897 (120579) = constant minus 119899 log120596 +

119899

sum

119894=1

log 119905 (119911119894

) +

119899

sum

119894=1

logΦ 119875119870

(119911119894

)

(3)And we have

120597119897 (120579)

120597120573= minus

1

120596

119899

sum

119894=1

(1199051015840

(119911119894

)

119905 (119911119894

)+ 1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

))

120597120578119894

120597120573

120597119897 (120579)

120597120596= minus

1

120596

119899

sum

119894=1

(1199051015840

(119911119894

)

119905 (119911119894

)119911119894

+ 119911119894

1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)+ 119899)

120597119897 (120579)

120597V=

119899

sum

119894=1

1199112

119894

119905 (119911119894

)+

119899

2(119863119866 (

V + 1

2) minus 119863119866 (

V2

) minus1

2)

120597119897 (120579)

1205971205722119895minus1

=

119899

sum

119894=1

1199112119895minus1

119894

120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)1198751015840

119870

(119911119894

) 119895 = 1 2 119870

(4)

where 120578119894

= 120578(x119894

120573) and 119863119866(119909) = (119889[log Γ(119909)])119889119909 The cor-responding second-order derivatives of (3) can be shown as

1205972

119897 (120579)

120597120573120597120573119879

=

119899

sum

119894=1

(120597120578119894

120597120573)

119879

1

1205962

[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

)

+ 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

120597120578119894

120597120573

minus1

120596[

1199051015840

(119911119894

)

119905 (119911119894

)+ 1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

1205972

120578119894

120597120573120597120573119879

1205972

119897 (120579)

120597120573120597120596=

119899

sum

119894=1

1

1205962

[1199051015840

(119911119894

)

119905 (119911119894

)+ 1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

+119911119894

1205962

[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

)

+ 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

120597120578119894

120597120573

1205972

119897 (120579)

120597120573120597V= minus

1

120596

119899

sum

119894=1

[

(1 + Vminus1) 1199113

119894

minus 119905 (119911119894

) 1199112

119894

V21199052 (119911119894

)]

120597120578119894

120597120573

1205972

119897 (120579)

1205971205731205971205722119895minus1

=

119899

sum

119894=1

1199112119895minus1

119894

[1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

) + 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

times120597120578119894

120597120573 119895 = 1 2 119870

1205972

119897 (120579)

1205971205962

=1

1205962

119899

sum

119894=1

[1199051015840

(119911119894

)

119905 (119911119894

)+ 1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)] 3119911119894

+ 1199112

119894

[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

)

+ 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

1205972

119897 (120579)

120597120596120597V= minus

1

120596

119899

sum

119894=1

[

(1 + Vminus1) 1199113

119894

minus 119905 (119911119894

) 1199112

119894

V21199052 (119911119894

)]

1205972

119897 (120579)

1205971205961205971205722119895minus1

= minus

119899

sum

119894=1

[1

120596(1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

) + 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)) 119911119894

]

times 1199112119895minus1

119894

119895 = 1 2 119870

4 Abstract and Applied Analysis

000

005

010

015

020

025

030

x

Density

FSNSTNFSTN

minus2 0 2 4 6

(a)

x

FSNSTNFSTN

000

005

010

015

020

025

030

035

Density

minus2 0 2 4 6

(b)

x

FSNSTNFSTN

minus2 0

000

005

010

015

020

025

030

Density

2 4 6

(c)

x

FSNSTNFSTN

000

005

010

015

020

025

030

Density

minus2 0 2 4 6

(d)

Figure 1 ((a)ndash(d)) Density functions of FSN STN and FSTN distributions with four different situations

1205972

119897 (120579)

120597V2=

119899

2[119879119866 (

V + 1

2) minus 119879119866 (

V2

)] +

119899

sum

119894=1

log (119905 (119911119894

))

1205972

119897 (120579)

1205971205722119895minus1

1205971205722119896minus1

=

119899

sum

119894=1

119911119895+119896

119894

119876 119875119870

(119911119894

) 119895 119896 = 1 2 119870

1205972

119897 (120579)

120597V1205971205722119895minus1

= 0 119895 = 1 2 119870

(5)

where 119879119866(119909) = (1198892

[log Γ(119909)])1198891199092 ℎ(119909) = (119905

10158401015840

(119909)119905(119909) minus

[1199051015840

(119909)]2

)1199052

(119909) and 119876(119909) = (120601(119909)Φ(119909))[119909 + (120601(119909)Φ(119909))]Assume119880(120579) = (119880

119879

(120573) 119880(120596) 119880(V) 119880119879(120572))119879 is the gradi-ent or score vector and 119867(120579) is the Hessian matrix composedof the above second-order derivatives To obtain the MLestimate of 120579 the Newton-Raphson iteration algorithm isdefined by

(119896+1)

= (119896)

minus [119867(120579)]

minus1

120579=

120579

(119896) [119880(120579)]120579=

120579

(119896) (6)

Abstract and Applied Analysis 5

It is noted that the above iterative procedure is an unpar-titioned algorithm that is all the parameters includingnonlinear regression coefficients 120573 scale parameter 120596 andshape parameter 120572 as well as tail thickness parameter V areestimated simultaneously For our considered problem thereare at least two difficulties that may be encountered for (6)the first one is that once the number of the parameters to beestimated becomes large the corresponding computationalburden turns to be heavy with an unacceptable estimationerror and the second one is as follows when the strength ofnonlinearity of the link function 120578(sdot sdot) changes the iterativeprocessmay becomeunstable or even nonconvergent leadingto the poor estimation results Considering the needs ofpractical problems rather than the total parameter set weare usually interested in a proper subset of it To improvethe efficiency of the algorithm and to facilitate statisticalinference of the nonlinearmodels with FSTNdistribution weput forward the following profile likelihoodmethod based on(3) and (6)

4 Profile Likelihood Inference

41 Profile Estimation Algorithm Let 120579 = (120573119879

120574119879

) be a parti-tion of 120579 where 120573 is a parameter vector of nonlinear regres-sion an interest parameter and 120574 is a 119902-dimension (119902 = 119870+

2) nuisance parameter with 120574 = (120572 120596 V)119879 Similarly thepartition of 119867 and 119880 is given as 119867(120579) = (

11986711 11986712

11986721 11986722) where

11986711

= (1205972

119897(120579)120597120573120597120573119879

) 11986712

= (1205972

119897(120579)120597120573120597120574119879

) = (1205972

119897(120579)

120597120573120597120596 1205972

119897(120579)120597120573120597V 1205972119897(120579)1205971205731205971205721

1205972

119897(120579)1205971205731205971205722119896minus1

) 11986712

= [11986721

]119879 and 119867

22

= (1205972

119897(120579) 120597120574120597120574119879

) and the diagonalelements of 119867

22

are given by 1205972

119897(120579) 1205971205962

1205972

119897(120579)120597V21205972

119897(120579)1205971205722

1

1205972

119897(120579)1205971205722

2119896minus1

respectively and the off-diagonal elements in 119867

22

can be obtained similarly Let119880(120579) = (119880

1

1198802

) corresponding to the partition of 120579In the subsequent context we focus on the estimation of120579 based on profile likelihood method [18] Firstly suppose 120573is known and we rewrite the original likelihood function (3)as

119897 (120579) = 119897 (120573 120574) = 119897120573

(120574) (7)

where notation 119897120573

(120574) denotes that 120573 is fixed but 120574 varies Foreach 120573 to estimate 120574 we can obtain

120573

= argmax120574

119897120573

(120574) (8)

Alternatively to estimate 120573 we evaluate the maximum valueof 119897120573

(120574) over 120573

and have

119901

= argmax120573

119897120573

(120573

) = argmax120573

119897 (120573 120573

) (9)

where 119897(120573 120573

) and 119901

are referred to as the profile likelihoodfunction and the profile ML estimation respectively

Following [19] we define the profile Newton-Raphsoniteration formula as follows

(119896+1)

119901

= (119896)

119901

minus [119867120573|120574

]minus1

(119880120573|120574

)

(119896+1)

119901

= (119896)

119901

minus ((11986722

)minus1

1198801

+ (11986722

)minus1

11986721

[119867120573|120574

]minus1

(119880120573|120574

))

(10)

where 119880120573|120574

= 1198801

minus 11986712

(11986722

)minus1

1198802

119867120573|120574

= 11986711

minus

11986712

(11986722

)minus1

11986721

and all the matrices and the vectors on theright hand side of (10) are evaluated at

(119896)

119901

and (119896)119901

Note that both in (6) and in (10) the strength of

nonlinearity of link function 120578(sdot sdot) is reflected by 1205972

120578119894

120597120573120597120573119879

to large extents Therefore by examination of the expressionof 1205972

119897(120579)120597120573120597120573119879 we find that when the element of

100381610038161003816100381610038161003816100381610038161003816

120596 [1199051015840

(119911119894

)

119905 (119911119894

)+ 1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

1205972

120578119894

120597120573120597120573119879

100381610038161003816100381610038161003816100381610038161003816

(11)

is much less than the corresponding element of1003816100381610038161003816100381610038161003816100381610038161003816

120597120578119879

119894

120597120573[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

) + 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

times120597120578119894

120597120573

100381610038161003816100381610038161003816100381610038161003816

(12)

the following approximating result can be obtained as

119867lowast

11

=

119899

sum

119894=1

(120597120578119894

120597120573)

119879

1

1205962

[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

)

+ 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

120597120578119894

120597120573

(13)

And then the iteration formulas of 119880lowast

120573|120574

[119867lowast

120573|120574

]minus1 119901

and

lowast

119901

can also be obtained just as before The above estimationprocedure is referred to as profile modified Newton-Raphsoniteration algorithm

The stopping rule for the above algorithm can bepresented as that iteration proceeds until some distanceinvolving two successive evaluations of the profile likeli-hood 119897(120573

120573

) such as |119897((119896+1)

120573

(119896+1)) minus 119897((119896)

120573

(119896))| or

|119897((119896+1)

120573

(119896+1))119897((119896)

120573

(119896))minus1| is small enough for example120576 lt 10

minus4 is adopted in this paperThe asymptotic covariancematrix of theML estimates for

profile likelihood can be evaluated by inverting the expectedinformation matrix however it does not have a closedform expression the observed information matrix 119869(120573) =

minus(1205972

119897(120573 120573

)120597120573120597120573119879

) can be used as a replacement which isestimated by 119867

11

minus 11986712

(11986722

)minus1

11986721

where 11986711

11986712

and 11986722

can be obtained as similar as aboveThe choice of the initial values plays an important role in

nonlinear regression fitting in this paper the specific steps

6 Abstract and Applied Analysis

for choosing the starting values are implemented as follows

(i) compute the initial value (0)

based on the nonlinearregression model with standard normal assumption

(ii) with (0)

fixed compute the initial values 120596(0) and

120572(0) for the SN and finite mixture SN assumptions

respectively

In order to simplify the estimation of parameter V wehave fixed integral values for V from 2 to 40 by one choose thevalue of V that maximizes the profile likelihood as V(0) andthen the initial values of

(0)

120596(0) 120572(0) V(0) that are requiredin the estimation procedure are all obtained

42 Profile Confidence Estimation and Hypothesis Test Con-fidence interval and hypothesis test play an important role instatistical inference and in the subsequent content we willconsider the profile confidence estimation andhypothesis testfor the parameters of interest in nonlinear regressionmodelsSuppose 119869(120579) = 119864[minus119867(120579)] the following regular conditionsfor likelihood inference are assumed

(R-i) int(120597 log 119871(x 120579)120597120579) sdot 119871(x 120579)119889x = 0

(R-ii) (120597120597120579) int 119871(x 120579) = int(120597119871(x 120579)120597120579)119889x = 0

(R-iii) (120597120597120579) int 119892(x)119871(x 120579)119889x = int 119892(x)(120597119871(x 120579)120597120579)119889xwhere 119892(x) is arbitrary measurable function

(R-iv) 119864(120597 log 119871(x 120579)120597120579)2

gt 0

Apart from the above assumptions in this paper someadditional conditions are assumed to hold for that

(A-i) 120578(x119894

120573) is twice continuously differentiable withrespect to 120573 isin 119877

119901(A-ii) 119863 = (120597120578

1

120597120573 120597120578119899

120597120573)119879 is full rank for all 120573

(A-iii) assume 1205790

is the true value of 120579 then there exist aneighbour region 119873

0

of 1205790

and a constant 119888 such that120582min(119869(120579)) gt 119888 for any 120579 isin 119873

0

Under the above assumption following [20] we have

2 119897 (119901

120573119901) minus 119897 (120573

0

1205730

) 997888rarr 1205942

(119901) (14)

where 1205730

is the true value of 120573 and the convergence is underthe meaning of convergence in probability Based on theprofile likelihood theory the confidence region of 120573 with theconfidence level 100(1 minus 120572) is given by

119862 (119884) = 120573 2 [119897 (119901

120573119901) minus 119897 (120573

120573

)] le 1205942

(119901 120572) (15)

where 1205942

(119901 120572) denotes the upper 120572 percentile of chi-squaredistribution with 119901 degrees of freedom Furthermore thehypothesis test

1198670

120573 = 1205730

against 1198671

120573 = 1205730

(16)

can also be considered and the corresponding test statisticsare presented by 2[119897(

120573

) minus 119897(1205730

1205730

)] Unlike standard

likelihood method profile likelihood confidence intervalsand hypothesis test do not need assumption of normality ofthe estimator or distribution which is too restrictive they arebased on an asymptotic chi-square distribution of the log pro-file likelihood ratio test statistics and these properties bring alot of convenience and feasibility in practical computation

5 Simulation Study

To investigate the experimental performance of our method-ology we undertake a simulation study to compare the fittingperformance of misspecified distribution as well as under-standing the large sample properties of the ML estimates Torealize this purpose we generate the artificial data from thefollowing two nonlinear models

Model (1) drug responsiveness model

119910119894

= 1205730

minus1205730

1 + (119909119894

1205731

)1205732

+ 120576119894

(17)

Model (2) nonlinear growth-curve model

119910119894

=1205731

1 + 1205732

exp (minus1205733

119909)+ 120576119894

(18)

with two kinds of skewed distributions for random error asfollows Case (I) 120576 sim STN(120583 120596 120572 V) and Case (II) 120576 sim

FSTN(120583 120596120572 V)The true parameter population is chosen as follows 120573 =

(2 5 minus08) is assumed to be the same for the above twomodels In Model (1) let 120583 = 120575

1

120596 = 15 120572 = 15 and V = 4

and in Model (2) let 120583 = 1205752

120596 = 15 120572 = (1 minus08) and V = 4Note that for the values of the degrees of freedom V a relativevalue (V = 4) can yield a heavy-tailed distribution as we need

In this simulation there are totally four simulated datasets corresponding to two nonlinear models Model (1) andModel (2) along with two distributions Case (I) and Case(II) for random error Similar to previous analysis eachsimulated data set is fitted under STN and FSN as well asFSTN scenarios using three different estimation algorithmsTo shed light on the experimental performance of ourmethodology an interesting comparison can be made byexamining how often we can recognize the true model

Table 1 shows the absolute value of the average biasbetween the true and the estimated parameters and thepercentages of each model being ranked as the best modelbased on AIC criterion out of 500 replications are alsopresented in Table 1

By examination of Table 1 we can find the following(i) the difference for the estimation results between drugresponsiveness model and growth-curve model is significantindicating that the nonlinearity of model imposes an impacton parameter estimation (ii) when the true distribution forrandom error is STN the three fitting distributions haveroughly the same behavior it is hard to tell which approachfor parameter estimation is good and which approach isbad and the similar results can be obtained for modelselection based on AIC (iii) when the true distribution forrandom error is FSTN the FSN and the FSTN distributions

Abstract and Applied Analysis 7

Table 1 Results of the simulation study

Method Model Error Fittingmodel

Bias and perc1205730

1205731

1205732

()

NR

1

Case ISTN 0048 0044 0034 384

FSN 0052 0043 0031 302

FSTN 0051 0048 0035 314

Case IISTN 0941 0445 0374 105

FSN 0035 0014 0011 438

FSTN 0033 0010 0010 457

2

Case ISTN 0067 0061 0053 362

FSN 0070 0057 0055 302

FSTN 0073 0072 0048 336

Case IISTN 0963 0469 0391 85

FSN 0047 0023 0017 446

FSTN 0043 0022 0021 469

P-NR

1

Case ISTN 0049 0040 0032 326

FSN 0050 0047 0038 401

FSTN 0052 0049 0036 273

Case IISTN 1029 0420 0248 114

FSN 0039 0018 0010 396

FSTN 0038 0017 0009 490

2

Case ISTN 0070 0050 0047 303

FSN 0067 0057 0056 423

FSTN 0073 0071 0050 274

Case IISTN 1052 0446 0265 102

FSN 0053 0031 0020 396

FSTN 0052 0022 0017 502

MP-NR

1

Case ISTN 0045 0049 0033 323FSN 0051 0047 0036 383FSTN 0045 0044 0031 294

Case IISTN 0773 0373 0266 145FSN 0031 0018 0011 384FSTN 0028 0010 0008 471

2

Case ISTN 0068 0072 0062 281FSN 0076 0074 0062 412FSTN 0073 0060 0054 307

Case IISTN 0794 0394 0290 113FSN 0037 0026 0017 394FSTN 0036 0021 0013 493

outperform the STNdistribution by producing estimateswithlower bias and higher AIC proportion furthermore the PNRand MPNR methods perform better than the traditional NRmethod in general

To study the consistence properties of ML estimate wefocus on the situation that the true distribution for randomerror is Case (II) whereas the fitting distribution is FSTN tooFor this case two estimation algorithms PNR and MPNR

are adopted for parameter estimation and samples of differentsizes (119899 = 50 100 and 200) are generated from Models (1)

and (2) We compute the 95 confidence interval for theparameters of interest and the mean square error (MSE) fordifferent model where MSE(120573) = (1500) sum

500

119894=1

120573(119894)

minus 1205732

The length of the 95 confidence interval and theMSE resultsare summarized in Table 2 From Table 2 we can see thatthe length of the confidence interval and the MSE tend todecrease as the sample size increases as expected

Tables 1 and 2 show that in general FSTN distributionenjoys more robustness and flexibility in modeling data withskewness and heavy tails as well asmultimodality in compari-sonwith other skewed alternatives and the implementation ofMPNR method brings more accuracy and improvement formodel estimation in the context of nonlinear regression withthis new distribution

6 Conclusion

We have proposed a new skewed distribution based on thegeneral FSS distribution framework called the FSTN dis-tribution which is allowed to accommodate multimodalityasymmetry and heavy tails jointly to offer greater flexibilitythan SN and STN counterpartsMoreover we have developeda modified profile of Newton-Raphson iterative algorithmfor estimating the parameters of interest of nonlinear modelwith FSTN distribution and the interval estimation andhypothesis test in a profile likelihood paradigm are alsoconsidered

Numerical studies reveal that in the context of nonlinearregression analysis if the true distribution is STN or FSNwhereas the fitting distribution is FSTN the estimationresults are not influenced by this misspecification of distribu-tion assumptionHowever once the true distribution is FSTNwhile the fitting distribution is STN the estimation resultsappear to be somewhat disappointing which shows therobustness of FSTN distribution In general the combinationof FSTN distribution with MPNR method brings moreaccuracy and improvement on the estimation of nonlinearregression

So far the present methodology is limited to the completedata analysis the extensions of this paper include missingdata version as well as Bayesian analysis of this model whichwill be reported in another paper

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 11171105 and 61273021 andin part by the Natural Science Foundation Project of CQcstc2013jjB40008

8 Abstract and Applied Analysis

Table 2 The table shows the length of 95 confidence interval of parameters of interest and MSE of specified model

Model Sample size Profile-NR Modified profile-NR1205730

1205731

1205732

MSE 1205730

1205731

1205732

MSE

150 1015 0675 0374 0195 0959 0589 0341 0167100 0558 0365 0152 0051 0519 0254 0132 0073200 0172 0134 0076 0004 0120 0141 0081 0005

250 1258 0828 0449 0240 1152 0692 0390 0154100 0569 0448 0208 0099 0516 0427 0211 0022200 0232 0181 0122 0009 0214 0145 0120 0008

References

[1] G Verbeke and E Lesaffre ldquoA linear mixed-effects model withheterogeneity in the random-effects populationrdquo Journal of theAmerican Statistical Association vol 91 no 433 pp 217ndash2211996

[2] P GhoshMD Branco andH Chakraborty ldquoBivariate randomeffectmodel using skew-normal distributionwith application toHIV-RNArdquo Statistics in Medicine vol 26 no 6 pp 1255ndash12672007

[3] A Azzalini and M G Genton ldquoRobust likelihood methodsbased on the skew-t and related distributionsrdquo InternationalStatistical Review vol 76 no 1 pp 106ndash129 2008

[4] A Azzalini ldquoA class of distributions which includes the normalonesrdquo Scandinavian Journal of Statistics vol 12 no 2 pp 171ndash178 1985

[5] A Azzalini and A Capitanio ldquoDistributions generated byperturbation of symmetrywith emphasis on amultivariate skew119905-distributionrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 65 no 2 pp 367ndash389 2003

[6] H W Gomez O Venegas and H Bolfarine ldquoSkew-symmetricdistributions generated by the distribution function of thenormal distributionrdquo Environmetrics vol 18 no 4 pp 395ndash4072007

[7] A Azzalini and A Dalla Valle ldquoThe multivariate skew-normaldistributionrdquo Biometrika vol 83 no 4 pp 715ndash726 1996

[8] A Azzalini and A Capitanio ldquoStatistical applications of themultivariate skew normal distributionrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 61 no 3 pp579ndash602 1999

[9] Y Ma and M G Genton ldquoFlexible class of skew-symmetricdistributionsrdquo Scandinavian Journal of Statistics vol 31 no 3pp 459ndash468 2004

[10] A Azzalini ldquoThe skew-normal distribution and related multi-variate familiesrdquo Scandinavian Journal of Statistics vol 32 no2 pp 159ndash200 2005

[11] H J Ho S Pyne and T I Lin ldquoMaximum likelihood inferencefor mixtures of skew Student-119905-normal distributions throughpractical EM-type algorithmsrdquo Statistics andComputing vol 22no 1 pp 287ndash299 2012

[12] C B Zeller VH Lachos and F E Vilca-Labra ldquoLocal influenceanalysis for regression models with scale mixtures of skew-normal distributionsrdquo Journal of Applied Statistics vol 38 no2 pp 343ndash368 2011

[13] V G Cancho D K Dey V H Lachos and M G AndradeldquoBayesian nonlinear regression models with scale mixturesof skew-normal distributions estimation and case influencediagnosticsrdquo Computational Statistics amp Data Analysis vol 55no 1 pp 588ndash602 2011

[14] M D Branco and D K Dey ldquoA general class of multivariateskew-elliptical distributionsrdquo Journal of Multivariate Analysisvol 79 no 1 pp 99ndash113 2001

[15] F C Xie J G Lin and B CWei ldquoDiagnostics for skew-normalnonlinear regression models with AR(1) errorsrdquo ComputationalStatistics and Data Analysis vol 53 no 12 pp 4403ndash4416 2009

[16] F-C Xie B-C Wei and J-G Lin ldquoHomogeneity diagnosticsfor skew-normal nonlinear regressions modelsrdquo Statistics ampProbability Letters vol 79 no 6 pp 821ndash827 2009

[17] L H Vanegas and F J Cysneiros ldquoAssessment of diagnosticprocedures in symmetrical nonlinear regression modelsrdquo Com-putational Statistics amp Data Analysis vol 54 no 4 pp 1002ndash1016 2010

[18] T A Severini ldquoAn approximation to the modified profilelikelihood functionrdquo Biometrika vol 85 no 2 pp 403ndash4111998

[19] G K Smyth ldquoPartitioned algorithms for maximum likelihoodand other non-linear estimationrdquo Statistics and Computing vol6 no 3 pp 201ndash216 1996

[20] B Wei Exponential Family Nonlinear Models Springer Singa-pore 1998

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Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

the definition of STN distribution and as a consequence itshares analogous feature with these two distributions Forall of that the FSTN distribution presents some interestingand peculiar features and is able to regulate the density ina more flexible way To be particular except for modelingskewness and tail thickness the FSTN distribution allows formultimodality depending on the specific setting of 120572 For thepurpose of comparison and illustration we assume 119891

0

(119911) =

120601(119911) and 119866(119911) = Φ(119911) in FSS distribution which is denotedby FSN and then we set 119870 = 2 in 119875

119870

(119911) that is 119875119870

(119909) =

1205721

119909 + 1205723

1199093 in STN distribution and for this case different

selections of 1205721

and 1205723

determine whether the density isunimodal or bimodal Moreover the same assumption for 119866

and 119875 in FSTN distribution is madeFigure 1 displays the density functions of FSN and STN

as well as FSTN distributions with four different situationsconsidered namely 120572

1

= 1 1205723

= 0 and V = 10 1205721

= 1205723

= 1

and V = 6 1205721

= 1 1205723

= minus1 and V = 4 1205721

= minus11205723

= 1 and V = 4 respectively with 120583 = 1 120596 = 15 for allcases By examination of Figure 1 we can detect how thesethree densities change with different combinations of 120572 andV For instance in Figure 1(a) FSN STN and FSTN appearto be very close while STN and FSTN are heavy tailed to alittle extent In Figure 1(b) both FSN and FSTN are unimodalwhen 120572

1

and 1205723

keep the same sign and the ranking for thedegree of skewness is STN FSTN and FSN in turn that isthe STN and FSTN distributions have thicker tails comparedto FSN distribution With opposite sign of 120572

1

and 1205723

bothFSN and FSTN distributions are bimodal and highly skewedin Figures 1(c) and 1(d) moreover in the same directionFSTN has thicker tails than FSN distribution Our proposedFSTN distribution can be treated as a proper compromisebetween the FSS distribution and the STN distribution Itallows for a wider range of tail behavior compared to FSSdistributionwhereas it is able to accommodatemultimodalitywhich cannot be described by STN distribution From theapplied viewpoint the FSTN distribution is an appealingoption which can be expected to yield robust inferentialresults in the presence of outlying observations

3 Likelihood Inference

Consider 119899 independent observations satisfying a nonlinearregression model as [119910

119894

] sim FSTN(120583119894

120596120572 V) with 120583119894

= 120578(x119894

120573) for 119894 = 1 2 119899 Here x

119894

is a 119901-dimensional vector and 120573is a 119901 times 1 vector of parameters Also let X = (x

1

x119899

)119879 be

the 119899 times 119901 design matrix 120578(sdot sdot) is a known twice differentiablefunction Then the corresponding log-likelihood for param-eter 120579 = (120573120572 120596 V) is given by

119897 (120579) = constant minus 119899 log120596 +

119899

sum

119894=1

log 119905 (119911119894

) +

119899

sum

119894=1

logΦ 119875119870

(119911119894

)

(3)And we have

120597119897 (120579)

120597120573= minus

1

120596

119899

sum

119894=1

(1199051015840

(119911119894

)

119905 (119911119894

)+ 1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

))

120597120578119894

120597120573

120597119897 (120579)

120597120596= minus

1

120596

119899

sum

119894=1

(1199051015840

(119911119894

)

119905 (119911119894

)119911119894

+ 119911119894

1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)+ 119899)

120597119897 (120579)

120597V=

119899

sum

119894=1

1199112

119894

119905 (119911119894

)+

119899

2(119863119866 (

V + 1

2) minus 119863119866 (

V2

) minus1

2)

120597119897 (120579)

1205971205722119895minus1

=

119899

sum

119894=1

1199112119895minus1

119894

120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)1198751015840

119870

(119911119894

) 119895 = 1 2 119870

(4)

where 120578119894

= 120578(x119894

120573) and 119863119866(119909) = (119889[log Γ(119909)])119889119909 The cor-responding second-order derivatives of (3) can be shown as

1205972

119897 (120579)

120597120573120597120573119879

=

119899

sum

119894=1

(120597120578119894

120597120573)

119879

1

1205962

[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

)

+ 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

120597120578119894

120597120573

minus1

120596[

1199051015840

(119911119894

)

119905 (119911119894

)+ 1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

1205972

120578119894

120597120573120597120573119879

1205972

119897 (120579)

120597120573120597120596=

119899

sum

119894=1

1

1205962

[1199051015840

(119911119894

)

119905 (119911119894

)+ 1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

+119911119894

1205962

[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

)

+ 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

120597120578119894

120597120573

1205972

119897 (120579)

120597120573120597V= minus

1

120596

119899

sum

119894=1

[

(1 + Vminus1) 1199113

119894

minus 119905 (119911119894

) 1199112

119894

V21199052 (119911119894

)]

120597120578119894

120597120573

1205972

119897 (120579)

1205971205731205971205722119895minus1

=

119899

sum

119894=1

1199112119895minus1

119894

[1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

) + 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

times120597120578119894

120597120573 119895 = 1 2 119870

1205972

119897 (120579)

1205971205962

=1

1205962

119899

sum

119894=1

[1199051015840

(119911119894

)

119905 (119911119894

)+ 1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)] 3119911119894

+ 1199112

119894

[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

)

+ 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

1205972

119897 (120579)

120597120596120597V= minus

1

120596

119899

sum

119894=1

[

(1 + Vminus1) 1199113

119894

minus 119905 (119911119894

) 1199112

119894

V21199052 (119911119894

)]

1205972

119897 (120579)

1205971205961205971205722119895minus1

= minus

119899

sum

119894=1

[1

120596(1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

) + 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)) 119911119894

]

times 1199112119895minus1

119894

119895 = 1 2 119870

4 Abstract and Applied Analysis

000

005

010

015

020

025

030

x

Density

FSNSTNFSTN

minus2 0 2 4 6

(a)

x

FSNSTNFSTN

000

005

010

015

020

025

030

035

Density

minus2 0 2 4 6

(b)

x

FSNSTNFSTN

minus2 0

000

005

010

015

020

025

030

Density

2 4 6

(c)

x

FSNSTNFSTN

000

005

010

015

020

025

030

Density

minus2 0 2 4 6

(d)

Figure 1 ((a)ndash(d)) Density functions of FSN STN and FSTN distributions with four different situations

1205972

119897 (120579)

120597V2=

119899

2[119879119866 (

V + 1

2) minus 119879119866 (

V2

)] +

119899

sum

119894=1

log (119905 (119911119894

))

1205972

119897 (120579)

1205971205722119895minus1

1205971205722119896minus1

=

119899

sum

119894=1

119911119895+119896

119894

119876 119875119870

(119911119894

) 119895 119896 = 1 2 119870

1205972

119897 (120579)

120597V1205971205722119895minus1

= 0 119895 = 1 2 119870

(5)

where 119879119866(119909) = (1198892

[log Γ(119909)])1198891199092 ℎ(119909) = (119905

10158401015840

(119909)119905(119909) minus

[1199051015840

(119909)]2

)1199052

(119909) and 119876(119909) = (120601(119909)Φ(119909))[119909 + (120601(119909)Φ(119909))]Assume119880(120579) = (119880

119879

(120573) 119880(120596) 119880(V) 119880119879(120572))119879 is the gradi-ent or score vector and 119867(120579) is the Hessian matrix composedof the above second-order derivatives To obtain the MLestimate of 120579 the Newton-Raphson iteration algorithm isdefined by

(119896+1)

= (119896)

minus [119867(120579)]

minus1

120579=

120579

(119896) [119880(120579)]120579=

120579

(119896) (6)

Abstract and Applied Analysis 5

It is noted that the above iterative procedure is an unpar-titioned algorithm that is all the parameters includingnonlinear regression coefficients 120573 scale parameter 120596 andshape parameter 120572 as well as tail thickness parameter V areestimated simultaneously For our considered problem thereare at least two difficulties that may be encountered for (6)the first one is that once the number of the parameters to beestimated becomes large the corresponding computationalburden turns to be heavy with an unacceptable estimationerror and the second one is as follows when the strength ofnonlinearity of the link function 120578(sdot sdot) changes the iterativeprocessmay becomeunstable or even nonconvergent leadingto the poor estimation results Considering the needs ofpractical problems rather than the total parameter set weare usually interested in a proper subset of it To improvethe efficiency of the algorithm and to facilitate statisticalinference of the nonlinearmodels with FSTNdistribution weput forward the following profile likelihoodmethod based on(3) and (6)

4 Profile Likelihood Inference

41 Profile Estimation Algorithm Let 120579 = (120573119879

120574119879

) be a parti-tion of 120579 where 120573 is a parameter vector of nonlinear regres-sion an interest parameter and 120574 is a 119902-dimension (119902 = 119870+

2) nuisance parameter with 120574 = (120572 120596 V)119879 Similarly thepartition of 119867 and 119880 is given as 119867(120579) = (

11986711 11986712

11986721 11986722) where

11986711

= (1205972

119897(120579)120597120573120597120573119879

) 11986712

= (1205972

119897(120579)120597120573120597120574119879

) = (1205972

119897(120579)

120597120573120597120596 1205972

119897(120579)120597120573120597V 1205972119897(120579)1205971205731205971205721

1205972

119897(120579)1205971205731205971205722119896minus1

) 11986712

= [11986721

]119879 and 119867

22

= (1205972

119897(120579) 120597120574120597120574119879

) and the diagonalelements of 119867

22

are given by 1205972

119897(120579) 1205971205962

1205972

119897(120579)120597V21205972

119897(120579)1205971205722

1

1205972

119897(120579)1205971205722

2119896minus1

respectively and the off-diagonal elements in 119867

22

can be obtained similarly Let119880(120579) = (119880

1

1198802

) corresponding to the partition of 120579In the subsequent context we focus on the estimation of120579 based on profile likelihood method [18] Firstly suppose 120573is known and we rewrite the original likelihood function (3)as

119897 (120579) = 119897 (120573 120574) = 119897120573

(120574) (7)

where notation 119897120573

(120574) denotes that 120573 is fixed but 120574 varies Foreach 120573 to estimate 120574 we can obtain

120573

= argmax120574

119897120573

(120574) (8)

Alternatively to estimate 120573 we evaluate the maximum valueof 119897120573

(120574) over 120573

and have

119901

= argmax120573

119897120573

(120573

) = argmax120573

119897 (120573 120573

) (9)

where 119897(120573 120573

) and 119901

are referred to as the profile likelihoodfunction and the profile ML estimation respectively

Following [19] we define the profile Newton-Raphsoniteration formula as follows

(119896+1)

119901

= (119896)

119901

minus [119867120573|120574

]minus1

(119880120573|120574

)

(119896+1)

119901

= (119896)

119901

minus ((11986722

)minus1

1198801

+ (11986722

)minus1

11986721

[119867120573|120574

]minus1

(119880120573|120574

))

(10)

where 119880120573|120574

= 1198801

minus 11986712

(11986722

)minus1

1198802

119867120573|120574

= 11986711

minus

11986712

(11986722

)minus1

11986721

and all the matrices and the vectors on theright hand side of (10) are evaluated at

(119896)

119901

and (119896)119901

Note that both in (6) and in (10) the strength of

nonlinearity of link function 120578(sdot sdot) is reflected by 1205972

120578119894

120597120573120597120573119879

to large extents Therefore by examination of the expressionof 1205972

119897(120579)120597120573120597120573119879 we find that when the element of

100381610038161003816100381610038161003816100381610038161003816

120596 [1199051015840

(119911119894

)

119905 (119911119894

)+ 1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

1205972

120578119894

120597120573120597120573119879

100381610038161003816100381610038161003816100381610038161003816

(11)

is much less than the corresponding element of1003816100381610038161003816100381610038161003816100381610038161003816

120597120578119879

119894

120597120573[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

) + 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

times120597120578119894

120597120573

100381610038161003816100381610038161003816100381610038161003816

(12)

the following approximating result can be obtained as

119867lowast

11

=

119899

sum

119894=1

(120597120578119894

120597120573)

119879

1

1205962

[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

)

+ 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

120597120578119894

120597120573

(13)

And then the iteration formulas of 119880lowast

120573|120574

[119867lowast

120573|120574

]minus1 119901

and

lowast

119901

can also be obtained just as before The above estimationprocedure is referred to as profile modified Newton-Raphsoniteration algorithm

The stopping rule for the above algorithm can bepresented as that iteration proceeds until some distanceinvolving two successive evaluations of the profile likeli-hood 119897(120573

120573

) such as |119897((119896+1)

120573

(119896+1)) minus 119897((119896)

120573

(119896))| or

|119897((119896+1)

120573

(119896+1))119897((119896)

120573

(119896))minus1| is small enough for example120576 lt 10

minus4 is adopted in this paperThe asymptotic covariancematrix of theML estimates for

profile likelihood can be evaluated by inverting the expectedinformation matrix however it does not have a closedform expression the observed information matrix 119869(120573) =

minus(1205972

119897(120573 120573

)120597120573120597120573119879

) can be used as a replacement which isestimated by 119867

11

minus 11986712

(11986722

)minus1

11986721

where 11986711

11986712

and 11986722

can be obtained as similar as aboveThe choice of the initial values plays an important role in

nonlinear regression fitting in this paper the specific steps

6 Abstract and Applied Analysis

for choosing the starting values are implemented as follows

(i) compute the initial value (0)

based on the nonlinearregression model with standard normal assumption

(ii) with (0)

fixed compute the initial values 120596(0) and

120572(0) for the SN and finite mixture SN assumptions

respectively

In order to simplify the estimation of parameter V wehave fixed integral values for V from 2 to 40 by one choose thevalue of V that maximizes the profile likelihood as V(0) andthen the initial values of

(0)

120596(0) 120572(0) V(0) that are requiredin the estimation procedure are all obtained

42 Profile Confidence Estimation and Hypothesis Test Con-fidence interval and hypothesis test play an important role instatistical inference and in the subsequent content we willconsider the profile confidence estimation andhypothesis testfor the parameters of interest in nonlinear regressionmodelsSuppose 119869(120579) = 119864[minus119867(120579)] the following regular conditionsfor likelihood inference are assumed

(R-i) int(120597 log 119871(x 120579)120597120579) sdot 119871(x 120579)119889x = 0

(R-ii) (120597120597120579) int 119871(x 120579) = int(120597119871(x 120579)120597120579)119889x = 0

(R-iii) (120597120597120579) int 119892(x)119871(x 120579)119889x = int 119892(x)(120597119871(x 120579)120597120579)119889xwhere 119892(x) is arbitrary measurable function

(R-iv) 119864(120597 log 119871(x 120579)120597120579)2

gt 0

Apart from the above assumptions in this paper someadditional conditions are assumed to hold for that

(A-i) 120578(x119894

120573) is twice continuously differentiable withrespect to 120573 isin 119877

119901(A-ii) 119863 = (120597120578

1

120597120573 120597120578119899

120597120573)119879 is full rank for all 120573

(A-iii) assume 1205790

is the true value of 120579 then there exist aneighbour region 119873

0

of 1205790

and a constant 119888 such that120582min(119869(120579)) gt 119888 for any 120579 isin 119873

0

Under the above assumption following [20] we have

2 119897 (119901

120573119901) minus 119897 (120573

0

1205730

) 997888rarr 1205942

(119901) (14)

where 1205730

is the true value of 120573 and the convergence is underthe meaning of convergence in probability Based on theprofile likelihood theory the confidence region of 120573 with theconfidence level 100(1 minus 120572) is given by

119862 (119884) = 120573 2 [119897 (119901

120573119901) minus 119897 (120573

120573

)] le 1205942

(119901 120572) (15)

where 1205942

(119901 120572) denotes the upper 120572 percentile of chi-squaredistribution with 119901 degrees of freedom Furthermore thehypothesis test

1198670

120573 = 1205730

against 1198671

120573 = 1205730

(16)

can also be considered and the corresponding test statisticsare presented by 2[119897(

120573

) minus 119897(1205730

1205730

)] Unlike standard

likelihood method profile likelihood confidence intervalsand hypothesis test do not need assumption of normality ofthe estimator or distribution which is too restrictive they arebased on an asymptotic chi-square distribution of the log pro-file likelihood ratio test statistics and these properties bring alot of convenience and feasibility in practical computation

5 Simulation Study

To investigate the experimental performance of our method-ology we undertake a simulation study to compare the fittingperformance of misspecified distribution as well as under-standing the large sample properties of the ML estimates Torealize this purpose we generate the artificial data from thefollowing two nonlinear models

Model (1) drug responsiveness model

119910119894

= 1205730

minus1205730

1 + (119909119894

1205731

)1205732

+ 120576119894

(17)

Model (2) nonlinear growth-curve model

119910119894

=1205731

1 + 1205732

exp (minus1205733

119909)+ 120576119894

(18)

with two kinds of skewed distributions for random error asfollows Case (I) 120576 sim STN(120583 120596 120572 V) and Case (II) 120576 sim

FSTN(120583 120596120572 V)The true parameter population is chosen as follows 120573 =

(2 5 minus08) is assumed to be the same for the above twomodels In Model (1) let 120583 = 120575

1

120596 = 15 120572 = 15 and V = 4

and in Model (2) let 120583 = 1205752

120596 = 15 120572 = (1 minus08) and V = 4Note that for the values of the degrees of freedom V a relativevalue (V = 4) can yield a heavy-tailed distribution as we need

In this simulation there are totally four simulated datasets corresponding to two nonlinear models Model (1) andModel (2) along with two distributions Case (I) and Case(II) for random error Similar to previous analysis eachsimulated data set is fitted under STN and FSN as well asFSTN scenarios using three different estimation algorithmsTo shed light on the experimental performance of ourmethodology an interesting comparison can be made byexamining how often we can recognize the true model

Table 1 shows the absolute value of the average biasbetween the true and the estimated parameters and thepercentages of each model being ranked as the best modelbased on AIC criterion out of 500 replications are alsopresented in Table 1

By examination of Table 1 we can find the following(i) the difference for the estimation results between drugresponsiveness model and growth-curve model is significantindicating that the nonlinearity of model imposes an impacton parameter estimation (ii) when the true distribution forrandom error is STN the three fitting distributions haveroughly the same behavior it is hard to tell which approachfor parameter estimation is good and which approach isbad and the similar results can be obtained for modelselection based on AIC (iii) when the true distribution forrandom error is FSTN the FSN and the FSTN distributions

Abstract and Applied Analysis 7

Table 1 Results of the simulation study

Method Model Error Fittingmodel

Bias and perc1205730

1205731

1205732

()

NR

1

Case ISTN 0048 0044 0034 384

FSN 0052 0043 0031 302

FSTN 0051 0048 0035 314

Case IISTN 0941 0445 0374 105

FSN 0035 0014 0011 438

FSTN 0033 0010 0010 457

2

Case ISTN 0067 0061 0053 362

FSN 0070 0057 0055 302

FSTN 0073 0072 0048 336

Case IISTN 0963 0469 0391 85

FSN 0047 0023 0017 446

FSTN 0043 0022 0021 469

P-NR

1

Case ISTN 0049 0040 0032 326

FSN 0050 0047 0038 401

FSTN 0052 0049 0036 273

Case IISTN 1029 0420 0248 114

FSN 0039 0018 0010 396

FSTN 0038 0017 0009 490

2

Case ISTN 0070 0050 0047 303

FSN 0067 0057 0056 423

FSTN 0073 0071 0050 274

Case IISTN 1052 0446 0265 102

FSN 0053 0031 0020 396

FSTN 0052 0022 0017 502

MP-NR

1

Case ISTN 0045 0049 0033 323FSN 0051 0047 0036 383FSTN 0045 0044 0031 294

Case IISTN 0773 0373 0266 145FSN 0031 0018 0011 384FSTN 0028 0010 0008 471

2

Case ISTN 0068 0072 0062 281FSN 0076 0074 0062 412FSTN 0073 0060 0054 307

Case IISTN 0794 0394 0290 113FSN 0037 0026 0017 394FSTN 0036 0021 0013 493

outperform the STNdistribution by producing estimateswithlower bias and higher AIC proportion furthermore the PNRand MPNR methods perform better than the traditional NRmethod in general

To study the consistence properties of ML estimate wefocus on the situation that the true distribution for randomerror is Case (II) whereas the fitting distribution is FSTN tooFor this case two estimation algorithms PNR and MPNR

are adopted for parameter estimation and samples of differentsizes (119899 = 50 100 and 200) are generated from Models (1)

and (2) We compute the 95 confidence interval for theparameters of interest and the mean square error (MSE) fordifferent model where MSE(120573) = (1500) sum

500

119894=1

120573(119894)

minus 1205732

The length of the 95 confidence interval and theMSE resultsare summarized in Table 2 From Table 2 we can see thatthe length of the confidence interval and the MSE tend todecrease as the sample size increases as expected

Tables 1 and 2 show that in general FSTN distributionenjoys more robustness and flexibility in modeling data withskewness and heavy tails as well asmultimodality in compari-sonwith other skewed alternatives and the implementation ofMPNR method brings more accuracy and improvement formodel estimation in the context of nonlinear regression withthis new distribution

6 Conclusion

We have proposed a new skewed distribution based on thegeneral FSS distribution framework called the FSTN dis-tribution which is allowed to accommodate multimodalityasymmetry and heavy tails jointly to offer greater flexibilitythan SN and STN counterpartsMoreover we have developeda modified profile of Newton-Raphson iterative algorithmfor estimating the parameters of interest of nonlinear modelwith FSTN distribution and the interval estimation andhypothesis test in a profile likelihood paradigm are alsoconsidered

Numerical studies reveal that in the context of nonlinearregression analysis if the true distribution is STN or FSNwhereas the fitting distribution is FSTN the estimationresults are not influenced by this misspecification of distribu-tion assumptionHowever once the true distribution is FSTNwhile the fitting distribution is STN the estimation resultsappear to be somewhat disappointing which shows therobustness of FSTN distribution In general the combinationof FSTN distribution with MPNR method brings moreaccuracy and improvement on the estimation of nonlinearregression

So far the present methodology is limited to the completedata analysis the extensions of this paper include missingdata version as well as Bayesian analysis of this model whichwill be reported in another paper

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 11171105 and 61273021 andin part by the Natural Science Foundation Project of CQcstc2013jjB40008

8 Abstract and Applied Analysis

Table 2 The table shows the length of 95 confidence interval of parameters of interest and MSE of specified model

Model Sample size Profile-NR Modified profile-NR1205730

1205731

1205732

MSE 1205730

1205731

1205732

MSE

150 1015 0675 0374 0195 0959 0589 0341 0167100 0558 0365 0152 0051 0519 0254 0132 0073200 0172 0134 0076 0004 0120 0141 0081 0005

250 1258 0828 0449 0240 1152 0692 0390 0154100 0569 0448 0208 0099 0516 0427 0211 0022200 0232 0181 0122 0009 0214 0145 0120 0008

References

[1] G Verbeke and E Lesaffre ldquoA linear mixed-effects model withheterogeneity in the random-effects populationrdquo Journal of theAmerican Statistical Association vol 91 no 433 pp 217ndash2211996

[2] P GhoshMD Branco andH Chakraborty ldquoBivariate randomeffectmodel using skew-normal distributionwith application toHIV-RNArdquo Statistics in Medicine vol 26 no 6 pp 1255ndash12672007

[3] A Azzalini and M G Genton ldquoRobust likelihood methodsbased on the skew-t and related distributionsrdquo InternationalStatistical Review vol 76 no 1 pp 106ndash129 2008

[4] A Azzalini ldquoA class of distributions which includes the normalonesrdquo Scandinavian Journal of Statistics vol 12 no 2 pp 171ndash178 1985

[5] A Azzalini and A Capitanio ldquoDistributions generated byperturbation of symmetrywith emphasis on amultivariate skew119905-distributionrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 65 no 2 pp 367ndash389 2003

[6] H W Gomez O Venegas and H Bolfarine ldquoSkew-symmetricdistributions generated by the distribution function of thenormal distributionrdquo Environmetrics vol 18 no 4 pp 395ndash4072007

[7] A Azzalini and A Dalla Valle ldquoThe multivariate skew-normaldistributionrdquo Biometrika vol 83 no 4 pp 715ndash726 1996

[8] A Azzalini and A Capitanio ldquoStatistical applications of themultivariate skew normal distributionrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 61 no 3 pp579ndash602 1999

[9] Y Ma and M G Genton ldquoFlexible class of skew-symmetricdistributionsrdquo Scandinavian Journal of Statistics vol 31 no 3pp 459ndash468 2004

[10] A Azzalini ldquoThe skew-normal distribution and related multi-variate familiesrdquo Scandinavian Journal of Statistics vol 32 no2 pp 159ndash200 2005

[11] H J Ho S Pyne and T I Lin ldquoMaximum likelihood inferencefor mixtures of skew Student-119905-normal distributions throughpractical EM-type algorithmsrdquo Statistics andComputing vol 22no 1 pp 287ndash299 2012

[12] C B Zeller VH Lachos and F E Vilca-Labra ldquoLocal influenceanalysis for regression models with scale mixtures of skew-normal distributionsrdquo Journal of Applied Statistics vol 38 no2 pp 343ndash368 2011

[13] V G Cancho D K Dey V H Lachos and M G AndradeldquoBayesian nonlinear regression models with scale mixturesof skew-normal distributions estimation and case influencediagnosticsrdquo Computational Statistics amp Data Analysis vol 55no 1 pp 588ndash602 2011

[14] M D Branco and D K Dey ldquoA general class of multivariateskew-elliptical distributionsrdquo Journal of Multivariate Analysisvol 79 no 1 pp 99ndash113 2001

[15] F C Xie J G Lin and B CWei ldquoDiagnostics for skew-normalnonlinear regression models with AR(1) errorsrdquo ComputationalStatistics and Data Analysis vol 53 no 12 pp 4403ndash4416 2009

[16] F-C Xie B-C Wei and J-G Lin ldquoHomogeneity diagnosticsfor skew-normal nonlinear regressions modelsrdquo Statistics ampProbability Letters vol 79 no 6 pp 821ndash827 2009

[17] L H Vanegas and F J Cysneiros ldquoAssessment of diagnosticprocedures in symmetrical nonlinear regression modelsrdquo Com-putational Statistics amp Data Analysis vol 54 no 4 pp 1002ndash1016 2010

[18] T A Severini ldquoAn approximation to the modified profilelikelihood functionrdquo Biometrika vol 85 no 2 pp 403ndash4111998

[19] G K Smyth ldquoPartitioned algorithms for maximum likelihoodand other non-linear estimationrdquo Statistics and Computing vol6 no 3 pp 201ndash216 1996

[20] B Wei Exponential Family Nonlinear Models Springer Singa-pore 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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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

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

4 Abstract and Applied Analysis

000

005

010

015

020

025

030

x

Density

FSNSTNFSTN

minus2 0 2 4 6

(a)

x

FSNSTNFSTN

000

005

010

015

020

025

030

035

Density

minus2 0 2 4 6

(b)

x

FSNSTNFSTN

minus2 0

000

005

010

015

020

025

030

Density

2 4 6

(c)

x

FSNSTNFSTN

000

005

010

015

020

025

030

Density

minus2 0 2 4 6

(d)

Figure 1 ((a)ndash(d)) Density functions of FSN STN and FSTN distributions with four different situations

1205972

119897 (120579)

120597V2=

119899

2[119879119866 (

V + 1

2) minus 119879119866 (

V2

)] +

119899

sum

119894=1

log (119905 (119911119894

))

1205972

119897 (120579)

1205971205722119895minus1

1205971205722119896minus1

=

119899

sum

119894=1

119911119895+119896

119894

119876 119875119870

(119911119894

) 119895 119896 = 1 2 119870

1205972

119897 (120579)

120597V1205971205722119895minus1

= 0 119895 = 1 2 119870

(5)

where 119879119866(119909) = (1198892

[log Γ(119909)])1198891199092 ℎ(119909) = (119905

10158401015840

(119909)119905(119909) minus

[1199051015840

(119909)]2

)1199052

(119909) and 119876(119909) = (120601(119909)Φ(119909))[119909 + (120601(119909)Φ(119909))]Assume119880(120579) = (119880

119879

(120573) 119880(120596) 119880(V) 119880119879(120572))119879 is the gradi-ent or score vector and 119867(120579) is the Hessian matrix composedof the above second-order derivatives To obtain the MLestimate of 120579 the Newton-Raphson iteration algorithm isdefined by

(119896+1)

= (119896)

minus [119867(120579)]

minus1

120579=

120579

(119896) [119880(120579)]120579=

120579

(119896) (6)

Abstract and Applied Analysis 5

It is noted that the above iterative procedure is an unpar-titioned algorithm that is all the parameters includingnonlinear regression coefficients 120573 scale parameter 120596 andshape parameter 120572 as well as tail thickness parameter V areestimated simultaneously For our considered problem thereare at least two difficulties that may be encountered for (6)the first one is that once the number of the parameters to beestimated becomes large the corresponding computationalburden turns to be heavy with an unacceptable estimationerror and the second one is as follows when the strength ofnonlinearity of the link function 120578(sdot sdot) changes the iterativeprocessmay becomeunstable or even nonconvergent leadingto the poor estimation results Considering the needs ofpractical problems rather than the total parameter set weare usually interested in a proper subset of it To improvethe efficiency of the algorithm and to facilitate statisticalinference of the nonlinearmodels with FSTNdistribution weput forward the following profile likelihoodmethod based on(3) and (6)

4 Profile Likelihood Inference

41 Profile Estimation Algorithm Let 120579 = (120573119879

120574119879

) be a parti-tion of 120579 where 120573 is a parameter vector of nonlinear regres-sion an interest parameter and 120574 is a 119902-dimension (119902 = 119870+

2) nuisance parameter with 120574 = (120572 120596 V)119879 Similarly thepartition of 119867 and 119880 is given as 119867(120579) = (

11986711 11986712

11986721 11986722) where

11986711

= (1205972

119897(120579)120597120573120597120573119879

) 11986712

= (1205972

119897(120579)120597120573120597120574119879

) = (1205972

119897(120579)

120597120573120597120596 1205972

119897(120579)120597120573120597V 1205972119897(120579)1205971205731205971205721

1205972

119897(120579)1205971205731205971205722119896minus1

) 11986712

= [11986721

]119879 and 119867

22

= (1205972

119897(120579) 120597120574120597120574119879

) and the diagonalelements of 119867

22

are given by 1205972

119897(120579) 1205971205962

1205972

119897(120579)120597V21205972

119897(120579)1205971205722

1

1205972

119897(120579)1205971205722

2119896minus1

respectively and the off-diagonal elements in 119867

22

can be obtained similarly Let119880(120579) = (119880

1

1198802

) corresponding to the partition of 120579In the subsequent context we focus on the estimation of120579 based on profile likelihood method [18] Firstly suppose 120573is known and we rewrite the original likelihood function (3)as

119897 (120579) = 119897 (120573 120574) = 119897120573

(120574) (7)

where notation 119897120573

(120574) denotes that 120573 is fixed but 120574 varies Foreach 120573 to estimate 120574 we can obtain

120573

= argmax120574

119897120573

(120574) (8)

Alternatively to estimate 120573 we evaluate the maximum valueof 119897120573

(120574) over 120573

and have

119901

= argmax120573

119897120573

(120573

) = argmax120573

119897 (120573 120573

) (9)

where 119897(120573 120573

) and 119901

are referred to as the profile likelihoodfunction and the profile ML estimation respectively

Following [19] we define the profile Newton-Raphsoniteration formula as follows

(119896+1)

119901

= (119896)

119901

minus [119867120573|120574

]minus1

(119880120573|120574

)

(119896+1)

119901

= (119896)

119901

minus ((11986722

)minus1

1198801

+ (11986722

)minus1

11986721

[119867120573|120574

]minus1

(119880120573|120574

))

(10)

where 119880120573|120574

= 1198801

minus 11986712

(11986722

)minus1

1198802

119867120573|120574

= 11986711

minus

11986712

(11986722

)minus1

11986721

and all the matrices and the vectors on theright hand side of (10) are evaluated at

(119896)

119901

and (119896)119901

Note that both in (6) and in (10) the strength of

nonlinearity of link function 120578(sdot sdot) is reflected by 1205972

120578119894

120597120573120597120573119879

to large extents Therefore by examination of the expressionof 1205972

119897(120579)120597120573120597120573119879 we find that when the element of

100381610038161003816100381610038161003816100381610038161003816

120596 [1199051015840

(119911119894

)

119905 (119911119894

)+ 1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

1205972

120578119894

120597120573120597120573119879

100381610038161003816100381610038161003816100381610038161003816

(11)

is much less than the corresponding element of1003816100381610038161003816100381610038161003816100381610038161003816

120597120578119879

119894

120597120573[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

) + 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

times120597120578119894

120597120573

100381610038161003816100381610038161003816100381610038161003816

(12)

the following approximating result can be obtained as

119867lowast

11

=

119899

sum

119894=1

(120597120578119894

120597120573)

119879

1

1205962

[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

)

+ 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

120597120578119894

120597120573

(13)

And then the iteration formulas of 119880lowast

120573|120574

[119867lowast

120573|120574

]minus1 119901

and

lowast

119901

can also be obtained just as before The above estimationprocedure is referred to as profile modified Newton-Raphsoniteration algorithm

The stopping rule for the above algorithm can bepresented as that iteration proceeds until some distanceinvolving two successive evaluations of the profile likeli-hood 119897(120573

120573

) such as |119897((119896+1)

120573

(119896+1)) minus 119897((119896)

120573

(119896))| or

|119897((119896+1)

120573

(119896+1))119897((119896)

120573

(119896))minus1| is small enough for example120576 lt 10

minus4 is adopted in this paperThe asymptotic covariancematrix of theML estimates for

profile likelihood can be evaluated by inverting the expectedinformation matrix however it does not have a closedform expression the observed information matrix 119869(120573) =

minus(1205972

119897(120573 120573

)120597120573120597120573119879

) can be used as a replacement which isestimated by 119867

11

minus 11986712

(11986722

)minus1

11986721

where 11986711

11986712

and 11986722

can be obtained as similar as aboveThe choice of the initial values plays an important role in

nonlinear regression fitting in this paper the specific steps

6 Abstract and Applied Analysis

for choosing the starting values are implemented as follows

(i) compute the initial value (0)

based on the nonlinearregression model with standard normal assumption

(ii) with (0)

fixed compute the initial values 120596(0) and

120572(0) for the SN and finite mixture SN assumptions

respectively

In order to simplify the estimation of parameter V wehave fixed integral values for V from 2 to 40 by one choose thevalue of V that maximizes the profile likelihood as V(0) andthen the initial values of

(0)

120596(0) 120572(0) V(0) that are requiredin the estimation procedure are all obtained

42 Profile Confidence Estimation and Hypothesis Test Con-fidence interval and hypothesis test play an important role instatistical inference and in the subsequent content we willconsider the profile confidence estimation andhypothesis testfor the parameters of interest in nonlinear regressionmodelsSuppose 119869(120579) = 119864[minus119867(120579)] the following regular conditionsfor likelihood inference are assumed

(R-i) int(120597 log 119871(x 120579)120597120579) sdot 119871(x 120579)119889x = 0

(R-ii) (120597120597120579) int 119871(x 120579) = int(120597119871(x 120579)120597120579)119889x = 0

(R-iii) (120597120597120579) int 119892(x)119871(x 120579)119889x = int 119892(x)(120597119871(x 120579)120597120579)119889xwhere 119892(x) is arbitrary measurable function

(R-iv) 119864(120597 log 119871(x 120579)120597120579)2

gt 0

Apart from the above assumptions in this paper someadditional conditions are assumed to hold for that

(A-i) 120578(x119894

120573) is twice continuously differentiable withrespect to 120573 isin 119877

119901(A-ii) 119863 = (120597120578

1

120597120573 120597120578119899

120597120573)119879 is full rank for all 120573

(A-iii) assume 1205790

is the true value of 120579 then there exist aneighbour region 119873

0

of 1205790

and a constant 119888 such that120582min(119869(120579)) gt 119888 for any 120579 isin 119873

0

Under the above assumption following [20] we have

2 119897 (119901

120573119901) minus 119897 (120573

0

1205730

) 997888rarr 1205942

(119901) (14)

where 1205730

is the true value of 120573 and the convergence is underthe meaning of convergence in probability Based on theprofile likelihood theory the confidence region of 120573 with theconfidence level 100(1 minus 120572) is given by

119862 (119884) = 120573 2 [119897 (119901

120573119901) minus 119897 (120573

120573

)] le 1205942

(119901 120572) (15)

where 1205942

(119901 120572) denotes the upper 120572 percentile of chi-squaredistribution with 119901 degrees of freedom Furthermore thehypothesis test

1198670

120573 = 1205730

against 1198671

120573 = 1205730

(16)

can also be considered and the corresponding test statisticsare presented by 2[119897(

120573

) minus 119897(1205730

1205730

)] Unlike standard

likelihood method profile likelihood confidence intervalsand hypothesis test do not need assumption of normality ofthe estimator or distribution which is too restrictive they arebased on an asymptotic chi-square distribution of the log pro-file likelihood ratio test statistics and these properties bring alot of convenience and feasibility in practical computation

5 Simulation Study

To investigate the experimental performance of our method-ology we undertake a simulation study to compare the fittingperformance of misspecified distribution as well as under-standing the large sample properties of the ML estimates Torealize this purpose we generate the artificial data from thefollowing two nonlinear models

Model (1) drug responsiveness model

119910119894

= 1205730

minus1205730

1 + (119909119894

1205731

)1205732

+ 120576119894

(17)

Model (2) nonlinear growth-curve model

119910119894

=1205731

1 + 1205732

exp (minus1205733

119909)+ 120576119894

(18)

with two kinds of skewed distributions for random error asfollows Case (I) 120576 sim STN(120583 120596 120572 V) and Case (II) 120576 sim

FSTN(120583 120596120572 V)The true parameter population is chosen as follows 120573 =

(2 5 minus08) is assumed to be the same for the above twomodels In Model (1) let 120583 = 120575

1

120596 = 15 120572 = 15 and V = 4

and in Model (2) let 120583 = 1205752

120596 = 15 120572 = (1 minus08) and V = 4Note that for the values of the degrees of freedom V a relativevalue (V = 4) can yield a heavy-tailed distribution as we need

In this simulation there are totally four simulated datasets corresponding to two nonlinear models Model (1) andModel (2) along with two distributions Case (I) and Case(II) for random error Similar to previous analysis eachsimulated data set is fitted under STN and FSN as well asFSTN scenarios using three different estimation algorithmsTo shed light on the experimental performance of ourmethodology an interesting comparison can be made byexamining how often we can recognize the true model

Table 1 shows the absolute value of the average biasbetween the true and the estimated parameters and thepercentages of each model being ranked as the best modelbased on AIC criterion out of 500 replications are alsopresented in Table 1

By examination of Table 1 we can find the following(i) the difference for the estimation results between drugresponsiveness model and growth-curve model is significantindicating that the nonlinearity of model imposes an impacton parameter estimation (ii) when the true distribution forrandom error is STN the three fitting distributions haveroughly the same behavior it is hard to tell which approachfor parameter estimation is good and which approach isbad and the similar results can be obtained for modelselection based on AIC (iii) when the true distribution forrandom error is FSTN the FSN and the FSTN distributions

Abstract and Applied Analysis 7

Table 1 Results of the simulation study

Method Model Error Fittingmodel

Bias and perc1205730

1205731

1205732

()

NR

1

Case ISTN 0048 0044 0034 384

FSN 0052 0043 0031 302

FSTN 0051 0048 0035 314

Case IISTN 0941 0445 0374 105

FSN 0035 0014 0011 438

FSTN 0033 0010 0010 457

2

Case ISTN 0067 0061 0053 362

FSN 0070 0057 0055 302

FSTN 0073 0072 0048 336

Case IISTN 0963 0469 0391 85

FSN 0047 0023 0017 446

FSTN 0043 0022 0021 469

P-NR

1

Case ISTN 0049 0040 0032 326

FSN 0050 0047 0038 401

FSTN 0052 0049 0036 273

Case IISTN 1029 0420 0248 114

FSN 0039 0018 0010 396

FSTN 0038 0017 0009 490

2

Case ISTN 0070 0050 0047 303

FSN 0067 0057 0056 423

FSTN 0073 0071 0050 274

Case IISTN 1052 0446 0265 102

FSN 0053 0031 0020 396

FSTN 0052 0022 0017 502

MP-NR

1

Case ISTN 0045 0049 0033 323FSN 0051 0047 0036 383FSTN 0045 0044 0031 294

Case IISTN 0773 0373 0266 145FSN 0031 0018 0011 384FSTN 0028 0010 0008 471

2

Case ISTN 0068 0072 0062 281FSN 0076 0074 0062 412FSTN 0073 0060 0054 307

Case IISTN 0794 0394 0290 113FSN 0037 0026 0017 394FSTN 0036 0021 0013 493

outperform the STNdistribution by producing estimateswithlower bias and higher AIC proportion furthermore the PNRand MPNR methods perform better than the traditional NRmethod in general

To study the consistence properties of ML estimate wefocus on the situation that the true distribution for randomerror is Case (II) whereas the fitting distribution is FSTN tooFor this case two estimation algorithms PNR and MPNR

are adopted for parameter estimation and samples of differentsizes (119899 = 50 100 and 200) are generated from Models (1)

and (2) We compute the 95 confidence interval for theparameters of interest and the mean square error (MSE) fordifferent model where MSE(120573) = (1500) sum

500

119894=1

120573(119894)

minus 1205732

The length of the 95 confidence interval and theMSE resultsare summarized in Table 2 From Table 2 we can see thatthe length of the confidence interval and the MSE tend todecrease as the sample size increases as expected

Tables 1 and 2 show that in general FSTN distributionenjoys more robustness and flexibility in modeling data withskewness and heavy tails as well asmultimodality in compari-sonwith other skewed alternatives and the implementation ofMPNR method brings more accuracy and improvement formodel estimation in the context of nonlinear regression withthis new distribution

6 Conclusion

We have proposed a new skewed distribution based on thegeneral FSS distribution framework called the FSTN dis-tribution which is allowed to accommodate multimodalityasymmetry and heavy tails jointly to offer greater flexibilitythan SN and STN counterpartsMoreover we have developeda modified profile of Newton-Raphson iterative algorithmfor estimating the parameters of interest of nonlinear modelwith FSTN distribution and the interval estimation andhypothesis test in a profile likelihood paradigm are alsoconsidered

Numerical studies reveal that in the context of nonlinearregression analysis if the true distribution is STN or FSNwhereas the fitting distribution is FSTN the estimationresults are not influenced by this misspecification of distribu-tion assumptionHowever once the true distribution is FSTNwhile the fitting distribution is STN the estimation resultsappear to be somewhat disappointing which shows therobustness of FSTN distribution In general the combinationof FSTN distribution with MPNR method brings moreaccuracy and improvement on the estimation of nonlinearregression

So far the present methodology is limited to the completedata analysis the extensions of this paper include missingdata version as well as Bayesian analysis of this model whichwill be reported in another paper

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 11171105 and 61273021 andin part by the Natural Science Foundation Project of CQcstc2013jjB40008

8 Abstract and Applied Analysis

Table 2 The table shows the length of 95 confidence interval of parameters of interest and MSE of specified model

Model Sample size Profile-NR Modified profile-NR1205730

1205731

1205732

MSE 1205730

1205731

1205732

MSE

150 1015 0675 0374 0195 0959 0589 0341 0167100 0558 0365 0152 0051 0519 0254 0132 0073200 0172 0134 0076 0004 0120 0141 0081 0005

250 1258 0828 0449 0240 1152 0692 0390 0154100 0569 0448 0208 0099 0516 0427 0211 0022200 0232 0181 0122 0009 0214 0145 0120 0008

References

[1] G Verbeke and E Lesaffre ldquoA linear mixed-effects model withheterogeneity in the random-effects populationrdquo Journal of theAmerican Statistical Association vol 91 no 433 pp 217ndash2211996

[2] P GhoshMD Branco andH Chakraborty ldquoBivariate randomeffectmodel using skew-normal distributionwith application toHIV-RNArdquo Statistics in Medicine vol 26 no 6 pp 1255ndash12672007

[3] A Azzalini and M G Genton ldquoRobust likelihood methodsbased on the skew-t and related distributionsrdquo InternationalStatistical Review vol 76 no 1 pp 106ndash129 2008

[4] A Azzalini ldquoA class of distributions which includes the normalonesrdquo Scandinavian Journal of Statistics vol 12 no 2 pp 171ndash178 1985

[5] A Azzalini and A Capitanio ldquoDistributions generated byperturbation of symmetrywith emphasis on amultivariate skew119905-distributionrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 65 no 2 pp 367ndash389 2003

[6] H W Gomez O Venegas and H Bolfarine ldquoSkew-symmetricdistributions generated by the distribution function of thenormal distributionrdquo Environmetrics vol 18 no 4 pp 395ndash4072007

[7] A Azzalini and A Dalla Valle ldquoThe multivariate skew-normaldistributionrdquo Biometrika vol 83 no 4 pp 715ndash726 1996

[8] A Azzalini and A Capitanio ldquoStatistical applications of themultivariate skew normal distributionrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 61 no 3 pp579ndash602 1999

[9] Y Ma and M G Genton ldquoFlexible class of skew-symmetricdistributionsrdquo Scandinavian Journal of Statistics vol 31 no 3pp 459ndash468 2004

[10] A Azzalini ldquoThe skew-normal distribution and related multi-variate familiesrdquo Scandinavian Journal of Statistics vol 32 no2 pp 159ndash200 2005

[11] H J Ho S Pyne and T I Lin ldquoMaximum likelihood inferencefor mixtures of skew Student-119905-normal distributions throughpractical EM-type algorithmsrdquo Statistics andComputing vol 22no 1 pp 287ndash299 2012

[12] C B Zeller VH Lachos and F E Vilca-Labra ldquoLocal influenceanalysis for regression models with scale mixtures of skew-normal distributionsrdquo Journal of Applied Statistics vol 38 no2 pp 343ndash368 2011

[13] V G Cancho D K Dey V H Lachos and M G AndradeldquoBayesian nonlinear regression models with scale mixturesof skew-normal distributions estimation and case influencediagnosticsrdquo Computational Statistics amp Data Analysis vol 55no 1 pp 588ndash602 2011

[14] M D Branco and D K Dey ldquoA general class of multivariateskew-elliptical distributionsrdquo Journal of Multivariate Analysisvol 79 no 1 pp 99ndash113 2001

[15] F C Xie J G Lin and B CWei ldquoDiagnostics for skew-normalnonlinear regression models with AR(1) errorsrdquo ComputationalStatistics and Data Analysis vol 53 no 12 pp 4403ndash4416 2009

[16] F-C Xie B-C Wei and J-G Lin ldquoHomogeneity diagnosticsfor skew-normal nonlinear regressions modelsrdquo Statistics ampProbability Letters vol 79 no 6 pp 821ndash827 2009

[17] L H Vanegas and F J Cysneiros ldquoAssessment of diagnosticprocedures in symmetrical nonlinear regression modelsrdquo Com-putational Statistics amp Data Analysis vol 54 no 4 pp 1002ndash1016 2010

[18] T A Severini ldquoAn approximation to the modified profilelikelihood functionrdquo Biometrika vol 85 no 2 pp 403ndash4111998

[19] G K Smyth ldquoPartitioned algorithms for maximum likelihoodand other non-linear estimationrdquo Statistics and Computing vol6 no 3 pp 201ndash216 1996

[20] B Wei Exponential Family Nonlinear Models Springer Singa-pore 1998

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

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OptimizationJournal of

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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

Abstract and Applied Analysis 5

It is noted that the above iterative procedure is an unpar-titioned algorithm that is all the parameters includingnonlinear regression coefficients 120573 scale parameter 120596 andshape parameter 120572 as well as tail thickness parameter V areestimated simultaneously For our considered problem thereare at least two difficulties that may be encountered for (6)the first one is that once the number of the parameters to beestimated becomes large the corresponding computationalburden turns to be heavy with an unacceptable estimationerror and the second one is as follows when the strength ofnonlinearity of the link function 120578(sdot sdot) changes the iterativeprocessmay becomeunstable or even nonconvergent leadingto the poor estimation results Considering the needs ofpractical problems rather than the total parameter set weare usually interested in a proper subset of it To improvethe efficiency of the algorithm and to facilitate statisticalinference of the nonlinearmodels with FSTNdistribution weput forward the following profile likelihoodmethod based on(3) and (6)

4 Profile Likelihood Inference

41 Profile Estimation Algorithm Let 120579 = (120573119879

120574119879

) be a parti-tion of 120579 where 120573 is a parameter vector of nonlinear regres-sion an interest parameter and 120574 is a 119902-dimension (119902 = 119870+

2) nuisance parameter with 120574 = (120572 120596 V)119879 Similarly thepartition of 119867 and 119880 is given as 119867(120579) = (

11986711 11986712

11986721 11986722) where

11986711

= (1205972

119897(120579)120597120573120597120573119879

) 11986712

= (1205972

119897(120579)120597120573120597120574119879

) = (1205972

119897(120579)

120597120573120597120596 1205972

119897(120579)120597120573120597V 1205972119897(120579)1205971205731205971205721

1205972

119897(120579)1205971205731205971205722119896minus1

) 11986712

= [11986721

]119879 and 119867

22

= (1205972

119897(120579) 120597120574120597120574119879

) and the diagonalelements of 119867

22

are given by 1205972

119897(120579) 1205971205962

1205972

119897(120579)120597V21205972

119897(120579)1205971205722

1

1205972

119897(120579)1205971205722

2119896minus1

respectively and the off-diagonal elements in 119867

22

can be obtained similarly Let119880(120579) = (119880

1

1198802

) corresponding to the partition of 120579In the subsequent context we focus on the estimation of120579 based on profile likelihood method [18] Firstly suppose 120573is known and we rewrite the original likelihood function (3)as

119897 (120579) = 119897 (120573 120574) = 119897120573

(120574) (7)

where notation 119897120573

(120574) denotes that 120573 is fixed but 120574 varies Foreach 120573 to estimate 120574 we can obtain

120573

= argmax120574

119897120573

(120574) (8)

Alternatively to estimate 120573 we evaluate the maximum valueof 119897120573

(120574) over 120573

and have

119901

= argmax120573

119897120573

(120573

) = argmax120573

119897 (120573 120573

) (9)

where 119897(120573 120573

) and 119901

are referred to as the profile likelihoodfunction and the profile ML estimation respectively

Following [19] we define the profile Newton-Raphsoniteration formula as follows

(119896+1)

119901

= (119896)

119901

minus [119867120573|120574

]minus1

(119880120573|120574

)

(119896+1)

119901

= (119896)

119901

minus ((11986722

)minus1

1198801

+ (11986722

)minus1

11986721

[119867120573|120574

]minus1

(119880120573|120574

))

(10)

where 119880120573|120574

= 1198801

minus 11986712

(11986722

)minus1

1198802

119867120573|120574

= 11986711

minus

11986712

(11986722

)minus1

11986721

and all the matrices and the vectors on theright hand side of (10) are evaluated at

(119896)

119901

and (119896)119901

Note that both in (6) and in (10) the strength of

nonlinearity of link function 120578(sdot sdot) is reflected by 1205972

120578119894

120597120573120597120573119879

to large extents Therefore by examination of the expressionof 1205972

119897(120579)120597120573120597120573119879 we find that when the element of

100381610038161003816100381610038161003816100381610038161003816

120596 [1199051015840

(119911119894

)

119905 (119911119894

)+ 1198751015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

1205972

120578119894

120597120573120597120573119879

100381610038161003816100381610038161003816100381610038161003816

(11)

is much less than the corresponding element of1003816100381610038161003816100381610038161003816100381610038161003816

120597120578119879

119894

120597120573[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

) + 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

times120597120578119894

120597120573

100381610038161003816100381610038161003816100381610038161003816

(12)

the following approximating result can be obtained as

119867lowast

11

=

119899

sum

119894=1

(120597120578119894

120597120573)

119879

1

1205962

[ℎ (119911119894

) + 1198751015840

119870

(119911119894

) 119876 119875119870

(119911119894

)

+ 11987510158401015840

119870

(119911119894

)120601 119875119870

(119911119894

)

Φ 119875119870

(119911119894

)]

120597120578119894

120597120573

(13)

And then the iteration formulas of 119880lowast

120573|120574

[119867lowast

120573|120574

]minus1 119901

and

lowast

119901

can also be obtained just as before The above estimationprocedure is referred to as profile modified Newton-Raphsoniteration algorithm

The stopping rule for the above algorithm can bepresented as that iteration proceeds until some distanceinvolving two successive evaluations of the profile likeli-hood 119897(120573

120573

) such as |119897((119896+1)

120573

(119896+1)) minus 119897((119896)

120573

(119896))| or

|119897((119896+1)

120573

(119896+1))119897((119896)

120573

(119896))minus1| is small enough for example120576 lt 10

minus4 is adopted in this paperThe asymptotic covariancematrix of theML estimates for

profile likelihood can be evaluated by inverting the expectedinformation matrix however it does not have a closedform expression the observed information matrix 119869(120573) =

minus(1205972

119897(120573 120573

)120597120573120597120573119879

) can be used as a replacement which isestimated by 119867

11

minus 11986712

(11986722

)minus1

11986721

where 11986711

11986712

and 11986722

can be obtained as similar as aboveThe choice of the initial values plays an important role in

nonlinear regression fitting in this paper the specific steps

6 Abstract and Applied Analysis

for choosing the starting values are implemented as follows

(i) compute the initial value (0)

based on the nonlinearregression model with standard normal assumption

(ii) with (0)

fixed compute the initial values 120596(0) and

120572(0) for the SN and finite mixture SN assumptions

respectively

In order to simplify the estimation of parameter V wehave fixed integral values for V from 2 to 40 by one choose thevalue of V that maximizes the profile likelihood as V(0) andthen the initial values of

(0)

120596(0) 120572(0) V(0) that are requiredin the estimation procedure are all obtained

42 Profile Confidence Estimation and Hypothesis Test Con-fidence interval and hypothesis test play an important role instatistical inference and in the subsequent content we willconsider the profile confidence estimation andhypothesis testfor the parameters of interest in nonlinear regressionmodelsSuppose 119869(120579) = 119864[minus119867(120579)] the following regular conditionsfor likelihood inference are assumed

(R-i) int(120597 log 119871(x 120579)120597120579) sdot 119871(x 120579)119889x = 0

(R-ii) (120597120597120579) int 119871(x 120579) = int(120597119871(x 120579)120597120579)119889x = 0

(R-iii) (120597120597120579) int 119892(x)119871(x 120579)119889x = int 119892(x)(120597119871(x 120579)120597120579)119889xwhere 119892(x) is arbitrary measurable function

(R-iv) 119864(120597 log 119871(x 120579)120597120579)2

gt 0

Apart from the above assumptions in this paper someadditional conditions are assumed to hold for that

(A-i) 120578(x119894

120573) is twice continuously differentiable withrespect to 120573 isin 119877

119901(A-ii) 119863 = (120597120578

1

120597120573 120597120578119899

120597120573)119879 is full rank for all 120573

(A-iii) assume 1205790

is the true value of 120579 then there exist aneighbour region 119873

0

of 1205790

and a constant 119888 such that120582min(119869(120579)) gt 119888 for any 120579 isin 119873

0

Under the above assumption following [20] we have

2 119897 (119901

120573119901) minus 119897 (120573

0

1205730

) 997888rarr 1205942

(119901) (14)

where 1205730

is the true value of 120573 and the convergence is underthe meaning of convergence in probability Based on theprofile likelihood theory the confidence region of 120573 with theconfidence level 100(1 minus 120572) is given by

119862 (119884) = 120573 2 [119897 (119901

120573119901) minus 119897 (120573

120573

)] le 1205942

(119901 120572) (15)

where 1205942

(119901 120572) denotes the upper 120572 percentile of chi-squaredistribution with 119901 degrees of freedom Furthermore thehypothesis test

1198670

120573 = 1205730

against 1198671

120573 = 1205730

(16)

can also be considered and the corresponding test statisticsare presented by 2[119897(

120573

) minus 119897(1205730

1205730

)] Unlike standard

likelihood method profile likelihood confidence intervalsand hypothesis test do not need assumption of normality ofthe estimator or distribution which is too restrictive they arebased on an asymptotic chi-square distribution of the log pro-file likelihood ratio test statistics and these properties bring alot of convenience and feasibility in practical computation

5 Simulation Study

To investigate the experimental performance of our method-ology we undertake a simulation study to compare the fittingperformance of misspecified distribution as well as under-standing the large sample properties of the ML estimates Torealize this purpose we generate the artificial data from thefollowing two nonlinear models

Model (1) drug responsiveness model

119910119894

= 1205730

minus1205730

1 + (119909119894

1205731

)1205732

+ 120576119894

(17)

Model (2) nonlinear growth-curve model

119910119894

=1205731

1 + 1205732

exp (minus1205733

119909)+ 120576119894

(18)

with two kinds of skewed distributions for random error asfollows Case (I) 120576 sim STN(120583 120596 120572 V) and Case (II) 120576 sim

FSTN(120583 120596120572 V)The true parameter population is chosen as follows 120573 =

(2 5 minus08) is assumed to be the same for the above twomodels In Model (1) let 120583 = 120575

1

120596 = 15 120572 = 15 and V = 4

and in Model (2) let 120583 = 1205752

120596 = 15 120572 = (1 minus08) and V = 4Note that for the values of the degrees of freedom V a relativevalue (V = 4) can yield a heavy-tailed distribution as we need

In this simulation there are totally four simulated datasets corresponding to two nonlinear models Model (1) andModel (2) along with two distributions Case (I) and Case(II) for random error Similar to previous analysis eachsimulated data set is fitted under STN and FSN as well asFSTN scenarios using three different estimation algorithmsTo shed light on the experimental performance of ourmethodology an interesting comparison can be made byexamining how often we can recognize the true model

Table 1 shows the absolute value of the average biasbetween the true and the estimated parameters and thepercentages of each model being ranked as the best modelbased on AIC criterion out of 500 replications are alsopresented in Table 1

By examination of Table 1 we can find the following(i) the difference for the estimation results between drugresponsiveness model and growth-curve model is significantindicating that the nonlinearity of model imposes an impacton parameter estimation (ii) when the true distribution forrandom error is STN the three fitting distributions haveroughly the same behavior it is hard to tell which approachfor parameter estimation is good and which approach isbad and the similar results can be obtained for modelselection based on AIC (iii) when the true distribution forrandom error is FSTN the FSN and the FSTN distributions

Abstract and Applied Analysis 7

Table 1 Results of the simulation study

Method Model Error Fittingmodel

Bias and perc1205730

1205731

1205732

()

NR

1

Case ISTN 0048 0044 0034 384

FSN 0052 0043 0031 302

FSTN 0051 0048 0035 314

Case IISTN 0941 0445 0374 105

FSN 0035 0014 0011 438

FSTN 0033 0010 0010 457

2

Case ISTN 0067 0061 0053 362

FSN 0070 0057 0055 302

FSTN 0073 0072 0048 336

Case IISTN 0963 0469 0391 85

FSN 0047 0023 0017 446

FSTN 0043 0022 0021 469

P-NR

1

Case ISTN 0049 0040 0032 326

FSN 0050 0047 0038 401

FSTN 0052 0049 0036 273

Case IISTN 1029 0420 0248 114

FSN 0039 0018 0010 396

FSTN 0038 0017 0009 490

2

Case ISTN 0070 0050 0047 303

FSN 0067 0057 0056 423

FSTN 0073 0071 0050 274

Case IISTN 1052 0446 0265 102

FSN 0053 0031 0020 396

FSTN 0052 0022 0017 502

MP-NR

1

Case ISTN 0045 0049 0033 323FSN 0051 0047 0036 383FSTN 0045 0044 0031 294

Case IISTN 0773 0373 0266 145FSN 0031 0018 0011 384FSTN 0028 0010 0008 471

2

Case ISTN 0068 0072 0062 281FSN 0076 0074 0062 412FSTN 0073 0060 0054 307

Case IISTN 0794 0394 0290 113FSN 0037 0026 0017 394FSTN 0036 0021 0013 493

outperform the STNdistribution by producing estimateswithlower bias and higher AIC proportion furthermore the PNRand MPNR methods perform better than the traditional NRmethod in general

To study the consistence properties of ML estimate wefocus on the situation that the true distribution for randomerror is Case (II) whereas the fitting distribution is FSTN tooFor this case two estimation algorithms PNR and MPNR

are adopted for parameter estimation and samples of differentsizes (119899 = 50 100 and 200) are generated from Models (1)

and (2) We compute the 95 confidence interval for theparameters of interest and the mean square error (MSE) fordifferent model where MSE(120573) = (1500) sum

500

119894=1

120573(119894)

minus 1205732

The length of the 95 confidence interval and theMSE resultsare summarized in Table 2 From Table 2 we can see thatthe length of the confidence interval and the MSE tend todecrease as the sample size increases as expected

Tables 1 and 2 show that in general FSTN distributionenjoys more robustness and flexibility in modeling data withskewness and heavy tails as well asmultimodality in compari-sonwith other skewed alternatives and the implementation ofMPNR method brings more accuracy and improvement formodel estimation in the context of nonlinear regression withthis new distribution

6 Conclusion

We have proposed a new skewed distribution based on thegeneral FSS distribution framework called the FSTN dis-tribution which is allowed to accommodate multimodalityasymmetry and heavy tails jointly to offer greater flexibilitythan SN and STN counterpartsMoreover we have developeda modified profile of Newton-Raphson iterative algorithmfor estimating the parameters of interest of nonlinear modelwith FSTN distribution and the interval estimation andhypothesis test in a profile likelihood paradigm are alsoconsidered

Numerical studies reveal that in the context of nonlinearregression analysis if the true distribution is STN or FSNwhereas the fitting distribution is FSTN the estimationresults are not influenced by this misspecification of distribu-tion assumptionHowever once the true distribution is FSTNwhile the fitting distribution is STN the estimation resultsappear to be somewhat disappointing which shows therobustness of FSTN distribution In general the combinationof FSTN distribution with MPNR method brings moreaccuracy and improvement on the estimation of nonlinearregression

So far the present methodology is limited to the completedata analysis the extensions of this paper include missingdata version as well as Bayesian analysis of this model whichwill be reported in another paper

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 11171105 and 61273021 andin part by the Natural Science Foundation Project of CQcstc2013jjB40008

8 Abstract and Applied Analysis

Table 2 The table shows the length of 95 confidence interval of parameters of interest and MSE of specified model

Model Sample size Profile-NR Modified profile-NR1205730

1205731

1205732

MSE 1205730

1205731

1205732

MSE

150 1015 0675 0374 0195 0959 0589 0341 0167100 0558 0365 0152 0051 0519 0254 0132 0073200 0172 0134 0076 0004 0120 0141 0081 0005

250 1258 0828 0449 0240 1152 0692 0390 0154100 0569 0448 0208 0099 0516 0427 0211 0022200 0232 0181 0122 0009 0214 0145 0120 0008

References

[1] G Verbeke and E Lesaffre ldquoA linear mixed-effects model withheterogeneity in the random-effects populationrdquo Journal of theAmerican Statistical Association vol 91 no 433 pp 217ndash2211996

[2] P GhoshMD Branco andH Chakraborty ldquoBivariate randomeffectmodel using skew-normal distributionwith application toHIV-RNArdquo Statistics in Medicine vol 26 no 6 pp 1255ndash12672007

[3] A Azzalini and M G Genton ldquoRobust likelihood methodsbased on the skew-t and related distributionsrdquo InternationalStatistical Review vol 76 no 1 pp 106ndash129 2008

[4] A Azzalini ldquoA class of distributions which includes the normalonesrdquo Scandinavian Journal of Statistics vol 12 no 2 pp 171ndash178 1985

[5] A Azzalini and A Capitanio ldquoDistributions generated byperturbation of symmetrywith emphasis on amultivariate skew119905-distributionrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 65 no 2 pp 367ndash389 2003

[6] H W Gomez O Venegas and H Bolfarine ldquoSkew-symmetricdistributions generated by the distribution function of thenormal distributionrdquo Environmetrics vol 18 no 4 pp 395ndash4072007

[7] A Azzalini and A Dalla Valle ldquoThe multivariate skew-normaldistributionrdquo Biometrika vol 83 no 4 pp 715ndash726 1996

[8] A Azzalini and A Capitanio ldquoStatistical applications of themultivariate skew normal distributionrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 61 no 3 pp579ndash602 1999

[9] Y Ma and M G Genton ldquoFlexible class of skew-symmetricdistributionsrdquo Scandinavian Journal of Statistics vol 31 no 3pp 459ndash468 2004

[10] A Azzalini ldquoThe skew-normal distribution and related multi-variate familiesrdquo Scandinavian Journal of Statistics vol 32 no2 pp 159ndash200 2005

[11] H J Ho S Pyne and T I Lin ldquoMaximum likelihood inferencefor mixtures of skew Student-119905-normal distributions throughpractical EM-type algorithmsrdquo Statistics andComputing vol 22no 1 pp 287ndash299 2012

[12] C B Zeller VH Lachos and F E Vilca-Labra ldquoLocal influenceanalysis for regression models with scale mixtures of skew-normal distributionsrdquo Journal of Applied Statistics vol 38 no2 pp 343ndash368 2011

[13] V G Cancho D K Dey V H Lachos and M G AndradeldquoBayesian nonlinear regression models with scale mixturesof skew-normal distributions estimation and case influencediagnosticsrdquo Computational Statistics amp Data Analysis vol 55no 1 pp 588ndash602 2011

[14] M D Branco and D K Dey ldquoA general class of multivariateskew-elliptical distributionsrdquo Journal of Multivariate Analysisvol 79 no 1 pp 99ndash113 2001

[15] F C Xie J G Lin and B CWei ldquoDiagnostics for skew-normalnonlinear regression models with AR(1) errorsrdquo ComputationalStatistics and Data Analysis vol 53 no 12 pp 4403ndash4416 2009

[16] F-C Xie B-C Wei and J-G Lin ldquoHomogeneity diagnosticsfor skew-normal nonlinear regressions modelsrdquo Statistics ampProbability Letters vol 79 no 6 pp 821ndash827 2009

[17] L H Vanegas and F J Cysneiros ldquoAssessment of diagnosticprocedures in symmetrical nonlinear regression modelsrdquo Com-putational Statistics amp Data Analysis vol 54 no 4 pp 1002ndash1016 2010

[18] T A Severini ldquoAn approximation to the modified profilelikelihood functionrdquo Biometrika vol 85 no 2 pp 403ndash4111998

[19] G K Smyth ldquoPartitioned algorithms for maximum likelihoodand other non-linear estimationrdquo Statistics and Computing vol6 no 3 pp 201ndash216 1996

[20] B Wei Exponential Family Nonlinear Models Springer Singa-pore 1998

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

6 Abstract and Applied Analysis

for choosing the starting values are implemented as follows

(i) compute the initial value (0)

based on the nonlinearregression model with standard normal assumption

(ii) with (0)

fixed compute the initial values 120596(0) and

120572(0) for the SN and finite mixture SN assumptions

respectively

In order to simplify the estimation of parameter V wehave fixed integral values for V from 2 to 40 by one choose thevalue of V that maximizes the profile likelihood as V(0) andthen the initial values of

(0)

120596(0) 120572(0) V(0) that are requiredin the estimation procedure are all obtained

42 Profile Confidence Estimation and Hypothesis Test Con-fidence interval and hypothesis test play an important role instatistical inference and in the subsequent content we willconsider the profile confidence estimation andhypothesis testfor the parameters of interest in nonlinear regressionmodelsSuppose 119869(120579) = 119864[minus119867(120579)] the following regular conditionsfor likelihood inference are assumed

(R-i) int(120597 log 119871(x 120579)120597120579) sdot 119871(x 120579)119889x = 0

(R-ii) (120597120597120579) int 119871(x 120579) = int(120597119871(x 120579)120597120579)119889x = 0

(R-iii) (120597120597120579) int 119892(x)119871(x 120579)119889x = int 119892(x)(120597119871(x 120579)120597120579)119889xwhere 119892(x) is arbitrary measurable function

(R-iv) 119864(120597 log 119871(x 120579)120597120579)2

gt 0

Apart from the above assumptions in this paper someadditional conditions are assumed to hold for that

(A-i) 120578(x119894

120573) is twice continuously differentiable withrespect to 120573 isin 119877

119901(A-ii) 119863 = (120597120578

1

120597120573 120597120578119899

120597120573)119879 is full rank for all 120573

(A-iii) assume 1205790

is the true value of 120579 then there exist aneighbour region 119873

0

of 1205790

and a constant 119888 such that120582min(119869(120579)) gt 119888 for any 120579 isin 119873

0

Under the above assumption following [20] we have

2 119897 (119901

120573119901) minus 119897 (120573

0

1205730

) 997888rarr 1205942

(119901) (14)

where 1205730

is the true value of 120573 and the convergence is underthe meaning of convergence in probability Based on theprofile likelihood theory the confidence region of 120573 with theconfidence level 100(1 minus 120572) is given by

119862 (119884) = 120573 2 [119897 (119901

120573119901) minus 119897 (120573

120573

)] le 1205942

(119901 120572) (15)

where 1205942

(119901 120572) denotes the upper 120572 percentile of chi-squaredistribution with 119901 degrees of freedom Furthermore thehypothesis test

1198670

120573 = 1205730

against 1198671

120573 = 1205730

(16)

can also be considered and the corresponding test statisticsare presented by 2[119897(

120573

) minus 119897(1205730

1205730

)] Unlike standard

likelihood method profile likelihood confidence intervalsand hypothesis test do not need assumption of normality ofthe estimator or distribution which is too restrictive they arebased on an asymptotic chi-square distribution of the log pro-file likelihood ratio test statistics and these properties bring alot of convenience and feasibility in practical computation

5 Simulation Study

To investigate the experimental performance of our method-ology we undertake a simulation study to compare the fittingperformance of misspecified distribution as well as under-standing the large sample properties of the ML estimates Torealize this purpose we generate the artificial data from thefollowing two nonlinear models

Model (1) drug responsiveness model

119910119894

= 1205730

minus1205730

1 + (119909119894

1205731

)1205732

+ 120576119894

(17)

Model (2) nonlinear growth-curve model

119910119894

=1205731

1 + 1205732

exp (minus1205733

119909)+ 120576119894

(18)

with two kinds of skewed distributions for random error asfollows Case (I) 120576 sim STN(120583 120596 120572 V) and Case (II) 120576 sim

FSTN(120583 120596120572 V)The true parameter population is chosen as follows 120573 =

(2 5 minus08) is assumed to be the same for the above twomodels In Model (1) let 120583 = 120575

1

120596 = 15 120572 = 15 and V = 4

and in Model (2) let 120583 = 1205752

120596 = 15 120572 = (1 minus08) and V = 4Note that for the values of the degrees of freedom V a relativevalue (V = 4) can yield a heavy-tailed distribution as we need

In this simulation there are totally four simulated datasets corresponding to two nonlinear models Model (1) andModel (2) along with two distributions Case (I) and Case(II) for random error Similar to previous analysis eachsimulated data set is fitted under STN and FSN as well asFSTN scenarios using three different estimation algorithmsTo shed light on the experimental performance of ourmethodology an interesting comparison can be made byexamining how often we can recognize the true model

Table 1 shows the absolute value of the average biasbetween the true and the estimated parameters and thepercentages of each model being ranked as the best modelbased on AIC criterion out of 500 replications are alsopresented in Table 1

By examination of Table 1 we can find the following(i) the difference for the estimation results between drugresponsiveness model and growth-curve model is significantindicating that the nonlinearity of model imposes an impacton parameter estimation (ii) when the true distribution forrandom error is STN the three fitting distributions haveroughly the same behavior it is hard to tell which approachfor parameter estimation is good and which approach isbad and the similar results can be obtained for modelselection based on AIC (iii) when the true distribution forrandom error is FSTN the FSN and the FSTN distributions

Abstract and Applied Analysis 7

Table 1 Results of the simulation study

Method Model Error Fittingmodel

Bias and perc1205730

1205731

1205732

()

NR

1

Case ISTN 0048 0044 0034 384

FSN 0052 0043 0031 302

FSTN 0051 0048 0035 314

Case IISTN 0941 0445 0374 105

FSN 0035 0014 0011 438

FSTN 0033 0010 0010 457

2

Case ISTN 0067 0061 0053 362

FSN 0070 0057 0055 302

FSTN 0073 0072 0048 336

Case IISTN 0963 0469 0391 85

FSN 0047 0023 0017 446

FSTN 0043 0022 0021 469

P-NR

1

Case ISTN 0049 0040 0032 326

FSN 0050 0047 0038 401

FSTN 0052 0049 0036 273

Case IISTN 1029 0420 0248 114

FSN 0039 0018 0010 396

FSTN 0038 0017 0009 490

2

Case ISTN 0070 0050 0047 303

FSN 0067 0057 0056 423

FSTN 0073 0071 0050 274

Case IISTN 1052 0446 0265 102

FSN 0053 0031 0020 396

FSTN 0052 0022 0017 502

MP-NR

1

Case ISTN 0045 0049 0033 323FSN 0051 0047 0036 383FSTN 0045 0044 0031 294

Case IISTN 0773 0373 0266 145FSN 0031 0018 0011 384FSTN 0028 0010 0008 471

2

Case ISTN 0068 0072 0062 281FSN 0076 0074 0062 412FSTN 0073 0060 0054 307

Case IISTN 0794 0394 0290 113FSN 0037 0026 0017 394FSTN 0036 0021 0013 493

outperform the STNdistribution by producing estimateswithlower bias and higher AIC proportion furthermore the PNRand MPNR methods perform better than the traditional NRmethod in general

To study the consistence properties of ML estimate wefocus on the situation that the true distribution for randomerror is Case (II) whereas the fitting distribution is FSTN tooFor this case two estimation algorithms PNR and MPNR

are adopted for parameter estimation and samples of differentsizes (119899 = 50 100 and 200) are generated from Models (1)

and (2) We compute the 95 confidence interval for theparameters of interest and the mean square error (MSE) fordifferent model where MSE(120573) = (1500) sum

500

119894=1

120573(119894)

minus 1205732

The length of the 95 confidence interval and theMSE resultsare summarized in Table 2 From Table 2 we can see thatthe length of the confidence interval and the MSE tend todecrease as the sample size increases as expected

Tables 1 and 2 show that in general FSTN distributionenjoys more robustness and flexibility in modeling data withskewness and heavy tails as well asmultimodality in compari-sonwith other skewed alternatives and the implementation ofMPNR method brings more accuracy and improvement formodel estimation in the context of nonlinear regression withthis new distribution

6 Conclusion

We have proposed a new skewed distribution based on thegeneral FSS distribution framework called the FSTN dis-tribution which is allowed to accommodate multimodalityasymmetry and heavy tails jointly to offer greater flexibilitythan SN and STN counterpartsMoreover we have developeda modified profile of Newton-Raphson iterative algorithmfor estimating the parameters of interest of nonlinear modelwith FSTN distribution and the interval estimation andhypothesis test in a profile likelihood paradigm are alsoconsidered

Numerical studies reveal that in the context of nonlinearregression analysis if the true distribution is STN or FSNwhereas the fitting distribution is FSTN the estimationresults are not influenced by this misspecification of distribu-tion assumptionHowever once the true distribution is FSTNwhile the fitting distribution is STN the estimation resultsappear to be somewhat disappointing which shows therobustness of FSTN distribution In general the combinationof FSTN distribution with MPNR method brings moreaccuracy and improvement on the estimation of nonlinearregression

So far the present methodology is limited to the completedata analysis the extensions of this paper include missingdata version as well as Bayesian analysis of this model whichwill be reported in another paper

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 11171105 and 61273021 andin part by the Natural Science Foundation Project of CQcstc2013jjB40008

8 Abstract and Applied Analysis

Table 2 The table shows the length of 95 confidence interval of parameters of interest and MSE of specified model

Model Sample size Profile-NR Modified profile-NR1205730

1205731

1205732

MSE 1205730

1205731

1205732

MSE

150 1015 0675 0374 0195 0959 0589 0341 0167100 0558 0365 0152 0051 0519 0254 0132 0073200 0172 0134 0076 0004 0120 0141 0081 0005

250 1258 0828 0449 0240 1152 0692 0390 0154100 0569 0448 0208 0099 0516 0427 0211 0022200 0232 0181 0122 0009 0214 0145 0120 0008

References

[1] G Verbeke and E Lesaffre ldquoA linear mixed-effects model withheterogeneity in the random-effects populationrdquo Journal of theAmerican Statistical Association vol 91 no 433 pp 217ndash2211996

[2] P GhoshMD Branco andH Chakraborty ldquoBivariate randomeffectmodel using skew-normal distributionwith application toHIV-RNArdquo Statistics in Medicine vol 26 no 6 pp 1255ndash12672007

[3] A Azzalini and M G Genton ldquoRobust likelihood methodsbased on the skew-t and related distributionsrdquo InternationalStatistical Review vol 76 no 1 pp 106ndash129 2008

[4] A Azzalini ldquoA class of distributions which includes the normalonesrdquo Scandinavian Journal of Statistics vol 12 no 2 pp 171ndash178 1985

[5] A Azzalini and A Capitanio ldquoDistributions generated byperturbation of symmetrywith emphasis on amultivariate skew119905-distributionrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 65 no 2 pp 367ndash389 2003

[6] H W Gomez O Venegas and H Bolfarine ldquoSkew-symmetricdistributions generated by the distribution function of thenormal distributionrdquo Environmetrics vol 18 no 4 pp 395ndash4072007

[7] A Azzalini and A Dalla Valle ldquoThe multivariate skew-normaldistributionrdquo Biometrika vol 83 no 4 pp 715ndash726 1996

[8] A Azzalini and A Capitanio ldquoStatistical applications of themultivariate skew normal distributionrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 61 no 3 pp579ndash602 1999

[9] Y Ma and M G Genton ldquoFlexible class of skew-symmetricdistributionsrdquo Scandinavian Journal of Statistics vol 31 no 3pp 459ndash468 2004

[10] A Azzalini ldquoThe skew-normal distribution and related multi-variate familiesrdquo Scandinavian Journal of Statistics vol 32 no2 pp 159ndash200 2005

[11] H J Ho S Pyne and T I Lin ldquoMaximum likelihood inferencefor mixtures of skew Student-119905-normal distributions throughpractical EM-type algorithmsrdquo Statistics andComputing vol 22no 1 pp 287ndash299 2012

[12] C B Zeller VH Lachos and F E Vilca-Labra ldquoLocal influenceanalysis for regression models with scale mixtures of skew-normal distributionsrdquo Journal of Applied Statistics vol 38 no2 pp 343ndash368 2011

[13] V G Cancho D K Dey V H Lachos and M G AndradeldquoBayesian nonlinear regression models with scale mixturesof skew-normal distributions estimation and case influencediagnosticsrdquo Computational Statistics amp Data Analysis vol 55no 1 pp 588ndash602 2011

[14] M D Branco and D K Dey ldquoA general class of multivariateskew-elliptical distributionsrdquo Journal of Multivariate Analysisvol 79 no 1 pp 99ndash113 2001

[15] F C Xie J G Lin and B CWei ldquoDiagnostics for skew-normalnonlinear regression models with AR(1) errorsrdquo ComputationalStatistics and Data Analysis vol 53 no 12 pp 4403ndash4416 2009

[16] F-C Xie B-C Wei and J-G Lin ldquoHomogeneity diagnosticsfor skew-normal nonlinear regressions modelsrdquo Statistics ampProbability Letters vol 79 no 6 pp 821ndash827 2009

[17] L H Vanegas and F J Cysneiros ldquoAssessment of diagnosticprocedures in symmetrical nonlinear regression modelsrdquo Com-putational Statistics amp Data Analysis vol 54 no 4 pp 1002ndash1016 2010

[18] T A Severini ldquoAn approximation to the modified profilelikelihood functionrdquo Biometrika vol 85 no 2 pp 403ndash4111998

[19] G K Smyth ldquoPartitioned algorithms for maximum likelihoodand other non-linear estimationrdquo Statistics and Computing vol6 no 3 pp 201ndash216 1996

[20] B Wei Exponential Family Nonlinear Models Springer Singa-pore 1998

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

Abstract and Applied Analysis 7

Table 1 Results of the simulation study

Method Model Error Fittingmodel

Bias and perc1205730

1205731

1205732

()

NR

1

Case ISTN 0048 0044 0034 384

FSN 0052 0043 0031 302

FSTN 0051 0048 0035 314

Case IISTN 0941 0445 0374 105

FSN 0035 0014 0011 438

FSTN 0033 0010 0010 457

2

Case ISTN 0067 0061 0053 362

FSN 0070 0057 0055 302

FSTN 0073 0072 0048 336

Case IISTN 0963 0469 0391 85

FSN 0047 0023 0017 446

FSTN 0043 0022 0021 469

P-NR

1

Case ISTN 0049 0040 0032 326

FSN 0050 0047 0038 401

FSTN 0052 0049 0036 273

Case IISTN 1029 0420 0248 114

FSN 0039 0018 0010 396

FSTN 0038 0017 0009 490

2

Case ISTN 0070 0050 0047 303

FSN 0067 0057 0056 423

FSTN 0073 0071 0050 274

Case IISTN 1052 0446 0265 102

FSN 0053 0031 0020 396

FSTN 0052 0022 0017 502

MP-NR

1

Case ISTN 0045 0049 0033 323FSN 0051 0047 0036 383FSTN 0045 0044 0031 294

Case IISTN 0773 0373 0266 145FSN 0031 0018 0011 384FSTN 0028 0010 0008 471

2

Case ISTN 0068 0072 0062 281FSN 0076 0074 0062 412FSTN 0073 0060 0054 307

Case IISTN 0794 0394 0290 113FSN 0037 0026 0017 394FSTN 0036 0021 0013 493

outperform the STNdistribution by producing estimateswithlower bias and higher AIC proportion furthermore the PNRand MPNR methods perform better than the traditional NRmethod in general

To study the consistence properties of ML estimate wefocus on the situation that the true distribution for randomerror is Case (II) whereas the fitting distribution is FSTN tooFor this case two estimation algorithms PNR and MPNR

are adopted for parameter estimation and samples of differentsizes (119899 = 50 100 and 200) are generated from Models (1)

and (2) We compute the 95 confidence interval for theparameters of interest and the mean square error (MSE) fordifferent model where MSE(120573) = (1500) sum

500

119894=1

120573(119894)

minus 1205732

The length of the 95 confidence interval and theMSE resultsare summarized in Table 2 From Table 2 we can see thatthe length of the confidence interval and the MSE tend todecrease as the sample size increases as expected

Tables 1 and 2 show that in general FSTN distributionenjoys more robustness and flexibility in modeling data withskewness and heavy tails as well asmultimodality in compari-sonwith other skewed alternatives and the implementation ofMPNR method brings more accuracy and improvement formodel estimation in the context of nonlinear regression withthis new distribution

6 Conclusion

We have proposed a new skewed distribution based on thegeneral FSS distribution framework called the FSTN dis-tribution which is allowed to accommodate multimodalityasymmetry and heavy tails jointly to offer greater flexibilitythan SN and STN counterpartsMoreover we have developeda modified profile of Newton-Raphson iterative algorithmfor estimating the parameters of interest of nonlinear modelwith FSTN distribution and the interval estimation andhypothesis test in a profile likelihood paradigm are alsoconsidered

Numerical studies reveal that in the context of nonlinearregression analysis if the true distribution is STN or FSNwhereas the fitting distribution is FSTN the estimationresults are not influenced by this misspecification of distribu-tion assumptionHowever once the true distribution is FSTNwhile the fitting distribution is STN the estimation resultsappear to be somewhat disappointing which shows therobustness of FSTN distribution In general the combinationof FSTN distribution with MPNR method brings moreaccuracy and improvement on the estimation of nonlinearregression

So far the present methodology is limited to the completedata analysis the extensions of this paper include missingdata version as well as Bayesian analysis of this model whichwill be reported in another paper

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 11171105 and 61273021 andin part by the Natural Science Foundation Project of CQcstc2013jjB40008

8 Abstract and Applied Analysis

Table 2 The table shows the length of 95 confidence interval of parameters of interest and MSE of specified model

Model Sample size Profile-NR Modified profile-NR1205730

1205731

1205732

MSE 1205730

1205731

1205732

MSE

150 1015 0675 0374 0195 0959 0589 0341 0167100 0558 0365 0152 0051 0519 0254 0132 0073200 0172 0134 0076 0004 0120 0141 0081 0005

250 1258 0828 0449 0240 1152 0692 0390 0154100 0569 0448 0208 0099 0516 0427 0211 0022200 0232 0181 0122 0009 0214 0145 0120 0008

References

[1] G Verbeke and E Lesaffre ldquoA linear mixed-effects model withheterogeneity in the random-effects populationrdquo Journal of theAmerican Statistical Association vol 91 no 433 pp 217ndash2211996

[2] P GhoshMD Branco andH Chakraborty ldquoBivariate randomeffectmodel using skew-normal distributionwith application toHIV-RNArdquo Statistics in Medicine vol 26 no 6 pp 1255ndash12672007

[3] A Azzalini and M G Genton ldquoRobust likelihood methodsbased on the skew-t and related distributionsrdquo InternationalStatistical Review vol 76 no 1 pp 106ndash129 2008

[4] A Azzalini ldquoA class of distributions which includes the normalonesrdquo Scandinavian Journal of Statistics vol 12 no 2 pp 171ndash178 1985

[5] A Azzalini and A Capitanio ldquoDistributions generated byperturbation of symmetrywith emphasis on amultivariate skew119905-distributionrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 65 no 2 pp 367ndash389 2003

[6] H W Gomez O Venegas and H Bolfarine ldquoSkew-symmetricdistributions generated by the distribution function of thenormal distributionrdquo Environmetrics vol 18 no 4 pp 395ndash4072007

[7] A Azzalini and A Dalla Valle ldquoThe multivariate skew-normaldistributionrdquo Biometrika vol 83 no 4 pp 715ndash726 1996

[8] A Azzalini and A Capitanio ldquoStatistical applications of themultivariate skew normal distributionrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 61 no 3 pp579ndash602 1999

[9] Y Ma and M G Genton ldquoFlexible class of skew-symmetricdistributionsrdquo Scandinavian Journal of Statistics vol 31 no 3pp 459ndash468 2004

[10] A Azzalini ldquoThe skew-normal distribution and related multi-variate familiesrdquo Scandinavian Journal of Statistics vol 32 no2 pp 159ndash200 2005

[11] H J Ho S Pyne and T I Lin ldquoMaximum likelihood inferencefor mixtures of skew Student-119905-normal distributions throughpractical EM-type algorithmsrdquo Statistics andComputing vol 22no 1 pp 287ndash299 2012

[12] C B Zeller VH Lachos and F E Vilca-Labra ldquoLocal influenceanalysis for regression models with scale mixtures of skew-normal distributionsrdquo Journal of Applied Statistics vol 38 no2 pp 343ndash368 2011

[13] V G Cancho D K Dey V H Lachos and M G AndradeldquoBayesian nonlinear regression models with scale mixturesof skew-normal distributions estimation and case influencediagnosticsrdquo Computational Statistics amp Data Analysis vol 55no 1 pp 588ndash602 2011

[14] M D Branco and D K Dey ldquoA general class of multivariateskew-elliptical distributionsrdquo Journal of Multivariate Analysisvol 79 no 1 pp 99ndash113 2001

[15] F C Xie J G Lin and B CWei ldquoDiagnostics for skew-normalnonlinear regression models with AR(1) errorsrdquo ComputationalStatistics and Data Analysis vol 53 no 12 pp 4403ndash4416 2009

[16] F-C Xie B-C Wei and J-G Lin ldquoHomogeneity diagnosticsfor skew-normal nonlinear regressions modelsrdquo Statistics ampProbability Letters vol 79 no 6 pp 821ndash827 2009

[17] L H Vanegas and F J Cysneiros ldquoAssessment of diagnosticprocedures in symmetrical nonlinear regression modelsrdquo Com-putational Statistics amp Data Analysis vol 54 no 4 pp 1002ndash1016 2010

[18] T A Severini ldquoAn approximation to the modified profilelikelihood functionrdquo Biometrika vol 85 no 2 pp 403ndash4111998

[19] G K Smyth ldquoPartitioned algorithms for maximum likelihoodand other non-linear estimationrdquo Statistics and Computing vol6 no 3 pp 201ndash216 1996

[20] B Wei Exponential Family Nonlinear Models Springer Singa-pore 1998

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 Abstract and Applied Analysis

Table 2 The table shows the length of 95 confidence interval of parameters of interest and MSE of specified model

Model Sample size Profile-NR Modified profile-NR1205730

1205731

1205732

MSE 1205730

1205731

1205732

MSE

150 1015 0675 0374 0195 0959 0589 0341 0167100 0558 0365 0152 0051 0519 0254 0132 0073200 0172 0134 0076 0004 0120 0141 0081 0005

250 1258 0828 0449 0240 1152 0692 0390 0154100 0569 0448 0208 0099 0516 0427 0211 0022200 0232 0181 0122 0009 0214 0145 0120 0008

References

[1] G Verbeke and E Lesaffre ldquoA linear mixed-effects model withheterogeneity in the random-effects populationrdquo Journal of theAmerican Statistical Association vol 91 no 433 pp 217ndash2211996

[2] P GhoshMD Branco andH Chakraborty ldquoBivariate randomeffectmodel using skew-normal distributionwith application toHIV-RNArdquo Statistics in Medicine vol 26 no 6 pp 1255ndash12672007

[3] A Azzalini and M G Genton ldquoRobust likelihood methodsbased on the skew-t and related distributionsrdquo InternationalStatistical Review vol 76 no 1 pp 106ndash129 2008

[4] A Azzalini ldquoA class of distributions which includes the normalonesrdquo Scandinavian Journal of Statistics vol 12 no 2 pp 171ndash178 1985

[5] A Azzalini and A Capitanio ldquoDistributions generated byperturbation of symmetrywith emphasis on amultivariate skew119905-distributionrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 65 no 2 pp 367ndash389 2003

[6] H W Gomez O Venegas and H Bolfarine ldquoSkew-symmetricdistributions generated by the distribution function of thenormal distributionrdquo Environmetrics vol 18 no 4 pp 395ndash4072007

[7] A Azzalini and A Dalla Valle ldquoThe multivariate skew-normaldistributionrdquo Biometrika vol 83 no 4 pp 715ndash726 1996

[8] A Azzalini and A Capitanio ldquoStatistical applications of themultivariate skew normal distributionrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 61 no 3 pp579ndash602 1999

[9] Y Ma and M G Genton ldquoFlexible class of skew-symmetricdistributionsrdquo Scandinavian Journal of Statistics vol 31 no 3pp 459ndash468 2004

[10] A Azzalini ldquoThe skew-normal distribution and related multi-variate familiesrdquo Scandinavian Journal of Statistics vol 32 no2 pp 159ndash200 2005

[11] H J Ho S Pyne and T I Lin ldquoMaximum likelihood inferencefor mixtures of skew Student-119905-normal distributions throughpractical EM-type algorithmsrdquo Statistics andComputing vol 22no 1 pp 287ndash299 2012

[12] C B Zeller VH Lachos and F E Vilca-Labra ldquoLocal influenceanalysis for regression models with scale mixtures of skew-normal distributionsrdquo Journal of Applied Statistics vol 38 no2 pp 343ndash368 2011

[13] V G Cancho D K Dey V H Lachos and M G AndradeldquoBayesian nonlinear regression models with scale mixturesof skew-normal distributions estimation and case influencediagnosticsrdquo Computational Statistics amp Data Analysis vol 55no 1 pp 588ndash602 2011

[14] M D Branco and D K Dey ldquoA general class of multivariateskew-elliptical distributionsrdquo Journal of Multivariate Analysisvol 79 no 1 pp 99ndash113 2001

[15] F C Xie J G Lin and B CWei ldquoDiagnostics for skew-normalnonlinear regression models with AR(1) errorsrdquo ComputationalStatistics and Data Analysis vol 53 no 12 pp 4403ndash4416 2009

[16] F-C Xie B-C Wei and J-G Lin ldquoHomogeneity diagnosticsfor skew-normal nonlinear regressions modelsrdquo Statistics ampProbability Letters vol 79 no 6 pp 821ndash827 2009

[17] L H Vanegas and F J Cysneiros ldquoAssessment of diagnosticprocedures in symmetrical nonlinear regression modelsrdquo Com-putational Statistics amp Data Analysis vol 54 no 4 pp 1002ndash1016 2010

[18] T A Severini ldquoAn approximation to the modified profilelikelihood functionrdquo Biometrika vol 85 no 2 pp 403ndash4111998

[19] G K Smyth ldquoPartitioned algorithms for maximum likelihoodand other non-linear estimationrdquo Statistics and Computing vol6 no 3 pp 201ndash216 1996

[20] B Wei Exponential Family Nonlinear Models Springer Singa-pore 1998

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