Research ArticleAsymptotic Optimality of Combined DoubleSequential Weighted Probability Ratio Test forThree Composite Hypotheses

Lei Wang Xiaolong Pu and Yan Li

School of Finance and Statistics East China Normal University Shanghai 200241 China

Correspondence should be addressed to Lei Wang leiwangstatgmailcom

Received 24 December 2014 Revised 13 March 2015 Accepted 15 March 2015

Academic Editor Antonino Laudani

Copyright copy 2015 Lei Wang et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We propose the weighted expected sample size (WESS) to evaluate the overall performance on the indifference-zones for threecomposite hypothesesrsquo testing problem Based on minimizing the WESS to control the expected sample sizes a new sequentialtest is developed by utilizing two double sequential weighted probability ratio tests (2-SWPRTs) simultaneously It is proven thatthe proposed test has a finite stopping time and is asymptotically optimal in the sense of asymptotically minimizing not only theexpected sample size but also any positive moment of the stopping time on the indifference-zones under some mild conditionsSimulation studies illustrate that the proposed test has the smallestWESS and relativemean index (RMI) comparedwith Sobel-Waldand Whitehead-Brunier tests

1 Introduction

Let 1198831 1198832 be independent and identically distributed

(iid) random variables whose common density function119891(119909 120579) (with respect to some nondegenerate measure ])belongs to the exponential family

119891 (119909 120579) = exp 119909120579 minus 120595 (120579) 120579 isin Θ = (120579 120579) (1)

where120595(sdot) is a convex function andΘ is the natural parameterspace with minusinfin le 120579 lt 120579 le infin The problem of interest is thefollowing three composite hypothesesrsquo testing problem

1198671 120579 le 120579

1versus

1198672 1205792le 120579 le 120579

3versus

1198673 120579 ge 120579

4

(1205791lt 1205792lt 1205793lt 1205794)

(2)

where 1205791 1205792 1205793 1205794isin Θ For example in clinical trial

applications in order to compare the effects of two drugs(Goeman et al [1]) the equivalence trial119867

0 |Δ| gt Δ

1versus

1198671 |Δ| lt Δ

0(0 lt Δ

0lt Δ1) would be more realistically

stated as 1198671 Δ lt minusΔ

1(inferiority) 119867

2 minusΔ0lt Δ lt Δ

0

(equivalence) and 1198673 Δ gt Δ

1(superiority) where Δ is

the difference of effect between two drugs The sequentialtesting of three or more hypotheses has been applied to avariety of engineering problems such as pattern recognition(Fu [2]McMillen andHolmes [3]) multiple-resolution radardetection (Bussgang [4]) products comparisons (Anderson[5]) and others (Li et al [6]) The intervals of [120579

1 1205792] and

[1205793 1205794] are usually called indifference-zones and denoted by

Θ = [1205791 1205792] cup [1205793 1205794]

Published work on this problem has taken two mainapproaches Pavlov [7] Baum and Veeravalli [8] and Dra-galin et al [9 10] studied the class of tests motivated by theBayesian framework The second approach has focused onextending the sequential probability ratio test (SPRT) anddouble sequential probability ratio test (2-SPRT) to incor-porate more than two hypotheses such as Sobel and Wald[11] Armitage [12] Simons [13] Lorden [14] Whitehead andBrunier [15] andLi andPu [16 17]Dragalin andNovikov [18]studied the problem of testing several composite hypotheseswith an indifference-zone for an unknown parameter Lai [19]considered the multihypothesis testing problem where someor all of these hypotheses are composite

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 356587 8 pageshttpdxdoiorg1011552015356587

2 Mathematical Problems in Engineering

Among others the tests proposed by Sobel and Wald[11] and Whitehead and Brunier [15] are usually used inpractice for problem (2) Specifically Sobel and Wald [11]proposed carrying out simultaneous SPRTs of 119867

1versus

1198672and 119867

2versus 119867

3 However when the true parameter

is in the indifference-zones the expected sample size ofthe Sobel-Wald test can be considerably larger than that ofa fixed-sample-size test plan Moreover it is untruncatedsuch that the number of observations required can not bepredetermined an undesirable property in many practicalsituations such as medical trial To reduce the maximumexpected sample size Whitehead and Brunier [15] appliedtwo 2-SPRTs instead of two SPRTs for the component testsat the cost of larger expected sample sizes when the trueparameter does not belong to the indifference-zones

For one-sided composite hypotheses in order to controlthe expected sample sizes Wang et al [20] proposed thedouble sequential weighted probability ratio test (2-SWPRT)based on mixture likelihood ratio statistics and showed thatthe 2-SWPRT is an asymptotically overall optimal test in thesense of asymptoticallyminimizing the expected sample sizeson the indifference-zone Motivated by the attractive proper-ties of the 2-SWPRT we extend the existing work on prob-lem (2) from pointwise optimality to overall performanceoptimality when there are different concerns of interest ondifferent 120579s In particular we propose an optimality criterionto evaluate the overall performance of sequential test planson the indifference-zones for three composite hypotheses andcorrespondingly develop a new sequential test for problem(2) by utilizing two 2-SWPRTs as the component tests toreduce the expected sample sizes We show the proposed testhas a finite stopping time and is asymptotically optimal inthe sense of asymptotically minimizing not only the expectedsample size but also any positive moment of the stoppingtime on the indifference-zones Simulation studies show thatthe proposed test not only has the smallest WESS comparedwith Sobel-Wald and Whitehead-Brunier tests but also issuperior to theWhitehead-Brunier test and comparable withthe Sobel-Wald test when the true parameter does not belongto the indifference-zones Moreover the RMI also shows theproposed test is an efficient method to improve the overallperformance

The rest of this paper is organized as follows In Section 2we review the Sobel-Wald and Whitehead-Brunier testsThe combined double sequential weighted probability ratiotest (denoted by combined 2-SWPRT) is proposed and itsproperties are given in Section 3 Simulation results areprovided in Section 4 and some conclusions are in Section 5All technical details are given in Appendix

2 Methodology Review

For one-sided composite hypotheses1198671versus119867

2 the SPRT

is optimal in the sense that it minimizes the expected samplesizes at 120579

1and 120579

2 and the 2-SPRT has (approximately)

minimal maximum expected sample size over (1205791 1205792) among

all sequential and nonsequential tests with the same errorprobabilities Given the well-known optimality properties ofthe SPRT and 2-SPRT it is natural to use the SPRTs and

2-SPRTs as the component tests to construct the sequentialtests for problem (2) respectively In this section we brieflyreview the Sobel-Wald and Whitehead-Brunier tests

For testing problem (2) the generalization of errors oftypes I and II is expressible in terms of a 3 times 3 error matrix119864 = (120572

119895119896) where 120572

119895119896= 119875[accepting 119867

119895| 119867119896is true] for

119895 119896 = 1 2 3 However under some mild conditions Sobeland Wald [11 pages 504-505] and Armitage [12 pages 142-143] showed that120572

31and12057213are zero which can be verified by

the simulation results in Section 4 It becomes apparent thatin the general case we have at most four ldquodegrees of freedomrdquoin choosing an error matrix Without loss of generality weconsiderΔ = (120591 119889) as a sequential test for problem (2) where120591 is the stopping rule and 119889 is the decision rule (119889 = 119896meansaccepting 119867

119896 119896 = 1 2 3) Set Θ

1= (120579 120579

1] Θ2= [1205792 1205793]

and Θ3= [1205794 120579) Given positive vectors 120572 = (120572

1 1205722) and

120573 = (1205731 1205732) (120572119894+ 120573119894lt 1 119894 = 1 2)

Υ (120572 120573) = Δ = (120591 119889) sup120579isinΘ119894

119875 (119889 = 119894 + 1) le 120572119894

sup120579isinΘ119894+1

119875 (119889 = 119894) le 120573119894 119894 = 1 2

(3)

is the set of all sequential tests with error probabilitiescontrolled by 120572 and 120573

(1) Sobel-Wald Test Since the hypotheses 1198671 1198672 and 119867

3

are ordered the sequential testing of problem (2) can beconstructed by combining the following two one-sided com-posite hypotheses 119878

1and 1198782

1198781 120579 le 120579

1versus 120579

2le 120579 le 120579

3

1198782 1205792le 120579 le 120579

3versus 120579 ge 120579

4

(4)

Sobel and Wald [11] proposed operating 1198781and 119878

2by the

SPRTs simultaneously For all 120579 120588 isin Θ define 119903119899(120579 120588) =

prod119899

119897=1119891(119909119897 120579)119891(119909

119897 120588) The stopping and decision rules of

119878119894(119894 = 1 2) determined by the SRPT are

120591119904

119894= inf 119899 ge 1 119903

119899(1205792119894 1205792119894minus1) notin (119860

119904

119894 119861119904

119894) 119894 = 1 2

119889119904

119894= 119868 (120591

119904

119894lt infin 119903

120591119904

119894

(1205792119894 1205792119894minus1) ge 119861119904

119894) + 119894

(5)

where 119868(sdot) is the indicator function and 119860119904119894and 119861119904

119894(119894 = 1 2)

are the boundary parameters (0 lt 119860119904119894lt 1 lt 119861

119904

119894lt infin) which

are usually set as

119860119904

119894=

120573119894

1 minus 120572119894

119861119904

119894=1 minus 120573119894

120572119894

119894 = 1 2 (6)

to meet requirements on the error probabilities When119861119904

1119861119904

2le 1 and 119860119904

1119860119904

2le 1 Sobel and Wald [11] showed

Mathematical Problems in Engineering 3

the event 1198891199041= 1 119889

119904

2= 3 is impossible The stopping and

decision rules of the Sobel-Wald test are defined as120591119904= max (120591119904

1 120591119904

2)

119889119904=

1 119889119904

1= 1 119889

119904

2= 2

2 119889119904

1= 2 119889

119904

2= 2

3 119889119904

1= 2 119889

119904

2= 3

(7)

The Sobel-Wald test is optimal in the sense that it minimizesthe expected sample sizes at 120579

2119894minus1and 1205792119894(119894 = 1 2) among all

sequential and nonsequential tests whose error probabilitiessatisfy Υ(120572 120573) However its expected sample sizes at otherparameters over Θmay be unsatisfactory

(2) Whitehead-Brunier Test In order to minimize the maxi-mum expected sample size under constraints (3) Whiteheadand Brunier [15] applied the 2-SPRT to operate 119878

1and

1198782 instead of the SPRT As in Lorden [21] let 119870(120579 120588) =

119864120579log[119891(119909 120579)119891(119909 120588)] be the Kullback-Leibler (KL) infor-

mation number Define 120579119894isin (1205792119894minus1 1205792119894) and 119899lowast

119894(119894 = 1 2) by

1003816100381610038161003816log1205721198941003816100381610038161003816

119870 (120579119894 1205792119894minus1)

=

1003816100381610038161003816log1205731198941003816100381610038161003816

119870 (120579119894 1205792119894)

= 119899lowast

119894 119894 = 1 2 (8)

Set 119888lowast119894such that Φ(119888lowast

119894) = minus119886

lowast

2119894(119886lowast

2119894minus1minus 119886lowast

2119894) 119894 = 1 2 where

Φ(sdot) is the cumulative distribution function of the standardnormal distribution 119886lowast

2119894minus1= (120579119894minus1205792119894minus1)119870(120579

119894 1205792119894minus1) and 119886lowast

2119894=

(120579119894minus 1205792119894)119870(120579

119894 1205792119894) 119894 = 1 2 Let

120579lowast

119894= 120579119894+ 119888lowast

119894[119899lowast

11989412059510158401015840(120579119894)]minus12

119894 = 1 2 (9)

The stopping and decision rules of 119878119894(119894 = 1 2) determined by

the 2-SPRT are120591119908

119894= inf 119899 ge 1 119903

119899(120579lowast

119894 1205792119894minus1) ge 119860119908

119894or 119903119899(120579lowast

119894 1205792119894) ge 119861119908

119894

119894 = 1 2

119889119908

119894= 119868 (120591

119908

119894lt infin 119903

120591119908

119894

(120579lowast

119894 1205792119894minus1) ge 119860119908

119894) + 119894

(10)

where 119860119908119894and 119861119908

119894(119894 = 1 2) are the boundary parameters

(0 lt 119860119908119894 119861119908

119894lt infin) The conservative values of 119860119908

119894and 119861119908

119894

are 1120572119894and 1120573

119894 in the sense that the real error probabilities

may be much smaller than 120572119894and 120573

119894(119894 = 1 2) respectively

The stopping and decision rules of the Whitehead-Bruniertest are defined as

120591119908= max (120591119908

1 120591119908

2)

119889119908=

1 119889119908

1= 1 119889

119908

2= 2

2 119889119908

1= 2 119889

119908

2= 2

3 119889119908

1= 2 119889

119908

2= 3

(11)

3 Optimality Criterion andCombined 2-SWPRT

For testing problem (2) if 120579 lt 1205791we prefer to accept 119867

1

and this preference is the stronger the smaller 120579 Similarly

if 120579 gt 1205794we prefer to accept 119867

3 and we prefer to accept

1198672if 1205792lt 120579 lt 120579

3 However we have no strong preference

between 1198671and 119867

2if 120579 isin [120579

1 1205792] and we also have no

strong preference between1198672and119867

3if 120579 isin [120579

3 1205794] In these

cases we need more observations for decision Thus whenthe error probabilities satisfy Υ(120572 120573) we focus on reductionof the expected sample sizes over the indifference-zonesΘ inapplications Let119892(120579) be a nonnegativeweight functionwhichis sectionally continuous on [120579

1 1205792] and [120579

3 1205794] respectively

and satisfies intΘ119892(120579)119889120579 = 1 We define the weighted expected

sample size as

WESS (119892) = intΘ

119864120579120591 sdot 119892 (120579) 119889120579 (12)

to evaluate the overall performance of sequential test plansonΘThe choice of 119892 should be chosen according to practicalneeds (Sobel andWald [11]) For example let 119892(120579) be uniformweights when there are no differences on Θ let 119892(120579) beassigned more weights when we focus more on reducing theexpected sample size on these parameter points As an overallevaluation theWESS(119892) integrates the performances onΘ byweighting the expected sample sizes

Motivated by Wang et al [20] we propose operating1198781and 119878

2by the 2-SWPRT Specifically the stopping and

decision rules of 119878119894(119894 = 1 2) by the 2-SWPRT are

120591lowast

119894= inf 119899 ge 1 119877119894

119899ge 119860119894or 119894119899ge 119861119894) 119894 = 1 2

119889119894= 119868 (120591

lowast

119894lt infin 119877

119894

120591lowast

119894

ge 119860119894) + 119894

(13)

where

119877119894

119899= int

1205792119894

120579lowast

119894

119903119899(120579 1205792119894minus1) 119892 (120579) 119889120579

119894

119899= int

120579lowast

119894

1205792119894minus1

119903119899(120579 1205792119894) 119892 (120579) 119889120579

119894 = 1 2

(14)

where 119860119894and 119861

119894(119894 = 1 2) are the boundary parameters (0 lt

119860119894 119861119894lt infin) Hence the stopping and decision rules of the

combined 2-SWPRT are defined as120591 = max (120591lowast

1 120591lowast

2)

119889 =

1 1198891= 1 119889

2= 2

2 1198891= 2 119889

2= 2

3 1198891= 2 119889

2= 3

(15)

Some features of the combined 2-SWPRT are providedin the following theorems whose proofs are provided inappendices

First we show the error probabilities of the combined2-SWPRT can be easily controlled and the stopping time isfinite

Theorem 1 There exist boundaries 119860119894and 119861

119894(119894 = 1 2) such

that (120591 119889) in (15) belongs to Υ(120572 120573)

4 Mathematical Problems in Engineering

Theorem 2 For any given nonnegative sectionally continuousweight function 119892(120579) the stopping time of the combined 2-SWPRT is finite

Second we prove that the combined 2-SWPRT is asymp-totically optimal on Θ

Definition 3 (120591 119889) isin Υ(120572 120573) is said to be asymptoticallyoptimal on Θ if

lim120572119894+120573119894rarr0

log(120572119894)asymplog(120573119894)

119864120579120591

minus log (120572119894+ 120573119894)= 119869119894(120579) 120579 isin [120579

2119894minus1 1205792119894] (16)

where 119869119894(120579) = min(1119870(120579 120579

2119894minus1) 1119870(120579 120579

2119894)) 119894 = 1 2

Theorem 4 When 119860119894= 120572minus1

119894and 119861

119894= 120573minus1

119894 the (120591 119889) defined

by (15) is asymptotically optimal on Θ

Third we show that any positive moment of the stoppingtime is asymptotically optimal on the indifference-zones

Theorem 5 Under the conditions of Theorem 4 for all 119902 ge 1and 120579 isin [120579

2119894minus1 1205792119894] 119894 = 1 2

lim120572119894+120573119894rarr0

log(120572119894)asymplog(120573119894)

119864120579[(

120591

minus log (120572119894+ 120573119894))

119902

] = (1

119869119894(120579))

119902

119894 = 1 2

(17)

4 Simulation Studies

In this section we conduct simulation studies to examinethe performances of the combined 2-SWPRT the Sobel-Wald test and Whitehead-Brunier test based on the normaland Bernoulli distributions In particular we considered twoweight functions for 119892(120579) as follows (1) uniform weights119892(120579) = 05sum

2

119894=1119868(120579 isin [120579

2119894minus1 1205792119894])(1205792119894minus1205792119894minus1) (2)KLweights

119892(120579) = 05sum2

119894=1119868(120579 isin [120579

2119894minus1 1205792119894])119872119894(120579)[int

1205792119894

1205792119894minus1

119872119894(120579)119889120579]

where119872119894(120579) = max(119870(120579 120579

2119894minus1) 119870(120579 120579

2119894)) As in Wang et al

[20] the corresponding formulations of the statistics 119877119894119899and

119894

119899(119894 = 1 2) can be obtained The boundaries of the tests

are determined through 106 Monte Carlo trials which makethe relative differences between the real error probabilities (1205721015840

119894

and 1205731015840119894) and the required ones (120572

119894and 120573

119894) within 1 that is

|1205721015840

119894minus 120572119894|120572119894lt 1 and |1205731015840

119894minus 120573119894|120573119894lt 1

Given the boundaries we obtained the simulatedWESS(119892) = sum

120579isin119878119864120579120591 sdot 119892(120579) to approximate integral (12) as

follows Let [1205791 1205792] and [120579

3 1205793] be discrete as the finite sets

of parameters 1198781= [1205791 1205791+ Δ 120579

1+ 2Δ 120579

lowast

1 120579

2minus Δ 120579

2]

and 1198782= [1205793 1205793+ Δ 120579

3+ 2Δ 120579

lowast

2 120579

4minus Δ 120579

4] with

increase Δ respectively Denote 119878 = 1198781cup 1198782sub Θ and the

weight function 119892(120579) is calculated based on 120579 isin 119878 thatis 119892(120579) = 05sum

2

119894=1119868(120579 isin 119878

119894)119872119894(120579)[sum

120579isin119878119894119872119894(120579)119889120579] for KL

weights We also compute the RMI to assess the relative

efficiency between different test plans According to Wang etal [20] we define

RMI (119892) = sum120579isin119878

119864120579120591 minus 119878119864

120579120591

119878119864120579120591

sdot 119892 (120579) (18)

where 119878119864120579120591 is the smallest119864

120579120591 among the compared tests that

is the Sobel-Wald test the Whitehead-Brunier test and thecombined 2-SWPRT A test plan with a smaller RMI(119892) valueis considered better in its overall performance

41 Test for the Normal Mean with Known Variance Suppose1198831 1198832 are iid from119873(120579 1) minus120579

1= 1205794= 15 minus120579

2= 1205793=

05 and 1205721= 1205722= 1205731= 1205732= 001 According to Lorden [21]

we have 120579lowast1= minus1 and 120579lowast

2= 1 The stopping boundaries are

obtained as follows(1) for the Sobel-Wald test 119860119904

119894= 0018 and 119861119904

119894= 5573

(2) for the Whitehead-Brunier test 119860119908119894= 119861119908

119894= 3736

(3) for the combined 2-SWPRT 119860119894= 119861119894= 2692 for the

uniform weights and 119860119894= 119861119894= 1378 for the KL

weights respectivelyAs expected we found that 120572

13and 12057231of these three tests are

equal to 0 Set Δ = 005 Through another simulation studywith 105 replications theWESS(119892) andRMI(119892) are presentedin Table 1 Similarly the expected sample sizes for 120579 isin [minus2 2]are illustrated in Figure 1

It is clear that the combined 2-SWPRTs have the smallestWESS(119892) in all cases In fact compared with the Sobel-Waldand Whitehead-Brunier tests the WESS(119892) of the combined2-SWPRT has been reduced by 1136 and 586 for theuniform weights and 813 and 757 for the KL weightsMeanwhile in terms of the RMI(119892) the combined 2-SWPRTalso performs best overall

FromFigure 1 it also can be seen that the expected samplesize of the combined 2-SWPRT is slightly larger than theWhitehead-Brunier test when the true parameter is close to120579lowast

119894(119894 = 1 2) and almost the same as the Sobel-Wald test when

the true parameter is close to 1205792119894minus1

or 1205792119894(119894 = 1 2) When

the true parameter belongs to Θ119896(119896 = 1 2 3) the combined

2-SWPRT performs better than the Whitehead-Brunier testand is comparable with the Sobel-Wald test

42 Test for the True Proportion of a Bernoulli DistributionSuppose 119883

1 1198832 are iid random variables from the

Bernoulli distribution and 119875(1198831= 1) = 119901 = 1 minus 119875(119883

1=

0) (0 lt 119901 lt 1) The three composite hypothesesrsquo testingproblem is

1198670 119901 le 119901

1versus

1198671 1199012le 119901 le 119901

3versus

1198673 119901 ge 119901

4

(19)

where 0 lt 1199011lt 1199012lt 1199013lt 1199014lt 1 Let 120572

1= 1205722= 1205731= 1205732=

001 1199011= 01 119901

2= 03 119901

3= 04 and 119901

4= 07 According to

(9) we have

119901lowast

119894=

log ((1 minus 1199012119894minus1) (1 minus 119901

2119894))

[log (11990121198941199012119894minus1) + log ((1 minus 119901

2119894minus1) (1 minus 119901

2119894))] (20)

Mathematical Problems in Engineering 5

Table 1 WESS(119892) and RMI(119892) for testing normal mean

119892Uniform weights KL weights

Sobel-Wald Whitehead-Brunier Combined 2-SWPRT Sobel-Wald Whitehead-Brunier Combined 2-SWPRTWESS 15231 14342 13511 13818 13734 12694RMI 0268 01652 00183 01629 02117 00178

Table 2 WESS(119892) and RMI(119892) for testing proportion in Bernoulli distribution

119892Uniform weights KL weights

Sobel-Wald Whitehead-Brunier Combined 2-SWPRT Sobel-Wald Whitehead-Brunier Combined 2-SWPRTWESS 48978 44163 43253 43387 41262 40035RMI 02508 00726 00163 01615 00985 00215

24

22

20

18

16

14

12

10

8

6

4

minus20 minus15 minus10 minus05 00 05 10 15 20

120579

Expe

cted

sam

ple s

ize

Sobel-WaldWhitehead-Brunier

Combined 2-SWPRT (uniform)Combined 2-SWPRT (KL)

Figure 1 Expected sample sizes for testing normal mean 1205721= 1205722=

1205731= 1205732= 001 minus120579

1= 1205794= 15 and minus120579

2= 1205793= 05

such that 119901lowast1= 0186 and 119901lowast

2= 0553 in the Whitehead-

Brunier test and combined 2-SWPRT The stopping bound-aries are obtained as follows

(1) for the Sobel-Wald test1198601199041= 0012 119861119904

1= 6652119860119904

2=

0014 and 1198611199042= 7723

(2) for the Whitehead-Brunier test 1198601199081= 3978 119861119908

1=

4651 1198601199082= 4433 and 119861119908

2= 4327

(3) for the combined 2-SWPRT 1198601= 1306 119861

1= 2032

1198602= 1696 and 119861

2= 1627 for the uniform weights

and 1198601= 1962 119861

1= 1435 119860

2= 1718 and 119861

2=

1327 for the KL weights

In this case the values of 12057213= 12057231= 0 Set Δ = 00625

Through another simulation study with 105 replications theWESS(119892) and RMI(119892) are presented in Table 2 Similarly theexpected sample sizes for 119901 isin (0 1) are illustrated in Figure 2

It can be seen from Table 2 that the combined 2-SWPRTstill has the smallest WESS(119892) and RMI(119892) for the Bernoullidistribution Meanwhile from Figure 2 we have similar

80

70

60

50

40

30

20

10

00 01 02 03 04 05 06 07 08 09 10

Expe

cted

sam

ple s

ize

Sobel-WaldWhitehead-Brunier

Combined 2-SWPRT (uniform)Combined 2-SWPRT (KL)

p

Figure 2 Expected sample sizes for testing proportion in Bernoullidistribution 120572

1= 1205722= 1205731= 1205732= 001 119901

2= 01 119901

2= 03 119901

3= 04

and 1199014= 07

conclusions as those in the normal distribution cases inSection 41

5 Summary

In this paper we propose theWESS(119892) to evaluate the overallperformance on the indifference-zones for three compositehypothesesrsquo testing problem In order to minimize WESS(119892)to control the expected sample sizes we developed a newsequential test by utilizing two 2-SWPRTs simultaneouslyWehave shown the proposed test is an asymptotically optimaltest in the sense of asymptotically minimizing the expectedsample sizes on the indifferent-zones

According to the simulation results compared with theSobel-Wald and Whitehead-Brunier tests we conclude thatthe proposed test has the following merits (1) it has thesmallest WESS(119892) and RMI(119892) (2) when the true parameteris close to 120579lowast

119894(119894 = 1 2) the proposed test has comparable

performance with Whitehead-Brunier test when the trueparameter is close to 120579

2119894minus1or 1205792119894(119894 = 1 2) it has almost

6 Mathematical Problems in Engineering

the same results as the Sobel-Wald test when the true param-eter does not belong to Θ the proposed test also performsbetter than the Whitehead-Brunier test and has comparableperformance with the Sobel-Wald test (3) the proposed testis easy to implement and can be extended to multihypothesistesting problems Future work will be concerned with themethod of determining the boundaries in an analytical wayinstead of the Monte Carlo method

Appendix

We provide sketch proofs of Theorems 1 2 4 and 5

Proof ofTheorem 1 LetF119899= 120590(119909

1 119909

119899) 119899 = 1 2 Note

that (1198771119899 119865119899 119899 ge 1) is a supermartingale under 119875

120579 forall120579 isin Θ

1

Therefore for all 120579 isin Θ1

119875120579(119889 = 2) le 119875

120579(1205911lt infin)

le int1205911ltinfin

119860minus1

11198771

1205911119889119875120579

le 119864120579[119860minus1

11198771

1]

(A1)

On the other hand following Lemma 1 of Chen and Hicker-nell [22] for any positive integer119898 and 120579 le 120579

1lt 120582 we have

119864120579[119903119898(120582 1205791)] le 1 (A2)

Thus

119864120579[119860minus1

11198771

1] = 119860

minus1

1int

1205792

120579lowast

1

119864120579[1199031(119905 1205791)] 119892 (119905) 119889119905

le 119860minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905

(A3)

Combining (A1) and (A3) we have

119875120579(119889 = 2) le 119860

minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905 forall120579 isin Θ1 (A4)

In particular setting

1198601= 120572minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905 (A5)

we have sup120579isinΘ1

119875120579(119889 = 2) le 120572

1 Similarly we can prove

that sup120579isinΘ2

119875120579(119889 = 1) le 120573

1 sup120579isinΘ2

119875120579(119889 = 3) le 120572

2 and

sup120579isinΘ3

119875120579(119889 = 2) le 120573

2with 119861

1= 120573minus1

1int120579lowast

1

1205791

119892(119905)119889119905 1198602=

120572minus1

2int1205794

120579lowast

2

119892(119905)119889119905 and 1198612= 120573minus1

2int120579lowast

2

1205793

119892(119905)119889119905 respectively

Proof of Theorem 2 If 119892(120579) is a sectionally continuous func-tion according to Theorem 32 of Wang et al [20] we knowthat (1) for all119860

1gt 0 and 119904

1gt (120595(120579

lowast

1)minus120595(120579

1))(120579lowast

1minus1205791) there

exists 1198791198601(1199041) lt infin such that 1198771

119899ge 1198601when 119899 ge 119879

1198601(1199041) and

119878119899ge 1198991199041 (2) for all 119861

1gt 0 and 119904

1lt (120595(120579

2)minus120595(120579

lowast

1))(1205792minus120579lowast

1)

there exists 1198791198611(1199041) lt infin such that 1

119899ge 1198611when 119899 ge 119879

1198611(1199041)

and 119878119899le 1198991199041 where 119878

119899= sum119899

119897=1119909119897

Noting that120595(120579) is convex we have (120595(120579lowast1)minus120595(120579

1))(120579lowast

1minus

1205791) lt (120595(120579

2) minus 120595(120579

lowast

1))(1205792minus 120579lowast

1) It is easy to choose 119904

1such

that

120595 (120579lowast

1) minus 120595 (120579

1)

120579lowast

1minus 1205791

lt 1199041lt120595 (1205792) minus 120595 (120579

lowast

1)

1205792minus 120579lowast

1

(A6)

Let 11987911986011198611

= max(1198791198601(1199041) 1198791198611(1199041)) Then we have 120591lowast

1le 11987911986011198611

Similarly for all 119860

2gt 0 and 119861

2gt 0 we can prove that there

exist 1198791198602(1199042) lt infin and 119879

1198612(1199042) lt infin such that 120591lowast

2le 11987911986021198612

Thus we have

120591 le max (11987911986011198611

11987911986021198612

) (A7)

Proof of Theorem 4 Using Hoeffding inequality (see Hoeffd-ing [23]) we know

lim inf1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)ge 1198691(120579) 120579 isin [120579

1 1205792] (A8)

so it suffices to show

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)le 1198691(120579) 120579 isin [120579

1 1205792] (A9)

According to Theorem 33 of Wang et al [20] for all 120579 isin

[120579lowast

1 1205792]

lim1205721rarr0

1205911

log (1198601)=

1

119870 (120579 1205791)

(as 119875120579) (A10)

Since 1198601= 120572minus1

1 when 120572

1+ 1205731rarr 0 and log(120572

1) asymp log(120573

1)

we have

minus log (1205721+ 1205731) 997888rarr minus log (120572

1) = log (119860

1) (A11)

Therefore for all 120579 isin [120579lowast1 1205792]

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591lowast

1

minus log (1205721+ 1205731)le lim1205721rarr0

1198641205791205911

log (1198601)=

1

119870 (120579 1205791)

(A12)

Similarly for all 120579 isin [1205791 120579lowast

1] we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591lowast

1

minus log (1205721+ 1205731)le lim1205731rarr0

1198641205791205911

log (1198611)=

1

119870 (120579 1205792)

(A13)

Combining two inequalities (A12) and (A13) we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)le 1198691(120579) 120579 isin [120579

1 1205792] (A14)

According to (A8) and (A14)

lim1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)= 1198691(120579) 120579 isin [120579

1 1205792] (A15)

Mathematical Problems in Engineering 7

Similarly we can prove that

lim1205722+1205732rarr0

log(1205722)asymplog(1205732)

119864120579120591

minus log (1205722+ 1205732)= 1198692(120579) 120579 isin [120579

3 1205794] (A16)

Proof of Theorem 5 Using Lemma 36 of Chen [24] for all119902 ge 1 we know

lim1205721rarr0

119864120579[(

1205911

minus log (1205721))

119902

] =1

[119870 (120579 1205791)]119902 (120579

lowast

1le 120579 le 120579

2)

lim1205731rarr0

119864120579[(

1205911

minus log (1205731))

119902

] =1

[119870 (120579 1205792)]119902 (120579

1le 120579 le 120579

lowast

1)

(A17)

Similar to Theorem 4 for all 120579 isin [120579lowast1 1205792] we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591lowast

1

minus log (1205721+ 1205731))]

119902

le lim1205721rarr0

119864120579[(

1205911

minus log (1205721))]

119902

=1

[119870 (120579 1205791)]119902

(A18)

For all 120579 isin [1205791 120579lowast

1] there is

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591lowast

1

minus log (1205721+ 1205731))]

119902

le lim1205731rarr0

119864120579[(

1205911

minus log (1205731))]

119902

=1

[119870 (120579 1205792)]119902

(A19)

According to (A18) (A19) and Hoeffding inequality wehave

lim1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591

minus log (1205721+ 1205731))

119902

]

= (1

1198691(120579))

119902

120579 isin [1205791 1205792]

(A20)

Similarly we can prove that

lim1205722+1205732rarr0

log(1205722)asymplog(1205732)

119864120579[(

120591

minus log (1205722+ 1205732))

119902

]

= (1

1198691(120579))

119902

120579 isin [1205793 1205794]

(A21)

Conflict of Interests

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

Acknowledgments

The authors would like to thank the Academic Edi-tor Antonino Laudani and an anonymous referee fortheir insightful comments and suggestions on this paperwhich have led to significant improvements This workwas supported by the Postdoctoral Science Foundation ofChina (2014M560317) the National Science Fund of China(11271135 11471119 11371142 and 11101156) the FundamentalResearch Funds for the Central Universities the 111 Project(B14019) and the Programof Shanghai Subject Chief Scientist(14XD1401600)

References

[1] J J Goeman A Solari and T Stijnen ldquoThree-sided hypothesistesting simultaneous testing of superiority equivalence andinferiorityrdquo Statistics in Medicine vol 29 no 20 pp 2117ndash21252010

[2] K S Fu Sequential Methods in Pattern Recognition and Learn-ing Academic Press New York NY USA 1968

[3] T McMillen and P Holmes ldquoThe dynamics of choice amongmultiple alternativesrdquo Journal of Mathematical Psychology vol50 no 1 pp 30ndash57 2006

[4] J J Bussgang ldquoSequential methods in radar detectionrdquo Proceed-ings of the IEEE vol 58 no 5 pp 731ndash743 1970

[5] S L Anderson ldquoSimple method of comparing the breakingstrength of two yarnsrdquo Journal of the Textle Institute vol 45 pp472ndash479 1954

[6] Y Li X L Pu and F Tsung ldquoAdaptive charting schemesbased on double sequential probability ratio testsrdquo Quality andReliability Engineering International vol 25 no 1 pp 21ndash392009

[7] I V Pavlov ldquoA sequential procedure for testingmany compositehypothesesrdquoTheory of Probability amp Its Applications vol 32 no1 pp 138ndash142 1988

[8] C W Baum and V V Veeravalli ldquoA sequential procedurefor multihypothesis testingrdquo IEEE Transactions on InformationTheory vol 40 no 6 pp 1994ndash2007 1994

[9] V P Dragalin A G Tartakovsky and V V Veeravalli ldquoMul-tihypothesis sequential probability ratio tests I asymptoticoptimalityrdquo IEEE Transactions on Information Theory vol 45no 7 pp 2448ndash2461 1999

[10] V P Dragalin A G Tartakovsky and V V Veeravalli ldquoMul-tihypothesis sequential probability ratio tests II Accurateasymptotic expansions for the expected sample sizerdquo IEEETransactions on Information Theory vol 46 no 4 pp 1366ndash1383 2000

[11] M Sobel and A Wald ldquoA sequential decision procedure forchoosing one of three hypotheses concerning the unknownmean of a normal distributionrdquo Annals of Mathematical Statis-tics vol 20 pp 502ndash522 1949

[12] P Armitage ldquoSequential analysis withmore than two alternativehypotheses and its relation to discriminant function analysisrdquoJournal of the Royal Statistical Society Series B Methodologicalvol 12 pp 137ndash144 1950

[13] G Simons ldquoLower bounds for average sample number ofsequential multihypothesis testsrdquo Annals of MathematicalStatistics vol 38 pp 1343ndash1364 1967

8 Mathematical Problems in Engineering

[14] G Lorden ldquoLikelihood ratio tests for sequential k-decisionproblemsrdquo Annals of Mathematical Statistics vol 43 pp 1412ndash1427 1972

[15] J Whitehead and H Brunier ldquoThe double triangular test asequential test for the two-sided alternative with early stoppingunder the null hypothesisrdquo Sequential Analysis Design Methodsamp Applications vol 9 no 2 pp 117ndash136 1990

[16] Y Li and X Pu ldquoHypothesis designs for three-hypothesis testproblemsrdquo Mathematical Problems in Engineering vol 2010Article ID 393095 15 pages 2010

[17] Y Li and X L Pu ldquoA method for designing three-hypothesistest problems and sequential schemesrdquo Communications inStatisticsmdashSimulation and Computation vol 39 no 9 pp 1690ndash1708 2010

[18] V P Dragalin and A A Novikov ldquoAdaptive sequential testsfor composite hypothesesrdquo Surveys of Applied and IndustrialMathematics vol 6 pp 387ndash398 1999

[19] T L Lai ldquoSequential multiple hypothesis testing and efficientfault detection-isolation in stochastic systemsrdquo IEEE Transac-tions on Information Theory vol 46 no 2 pp 595ndash608 2000

[20] L Wang D D Xiang X L Pu and Y Li ldquoA double sequen-tial weighted probability ratio test for one-sided compositehypothesesrdquo Communications in StatisticsTheory andMethodsvol 42 no 20 pp 3678ndash3695 2013

[21] G Lorden ldquo2-SPRTrsquos and the modified Kiefer-Weiss problemof minimizing an expected sample sizerdquoTheAnnals of Statisticsvol 4 no 2 pp 281ndash291 1976

[22] J D Chen and F J Hickernell ldquoA class of asymptotically optimalsequential tests for composite hypothesesrdquo Science in ChinaSeries A vol 37 no 11 pp 1314ndash1324 1994

[23] W Hoeffding ldquoLower bounds for the expected sample sizeand the average risk of a sequential procedurerdquo Annals ofMathematical Statistics vol 31 pp 352ndash368 1960

[24] J D Chen ldquoAsymptotic optimality for one class of truncatedsequential testsrdquo Science in China Series A MathematicsPhysics Astronomy vol 30 pp 30ndash41 1991

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

2 Mathematical Problems in Engineering

Among others the tests proposed by Sobel and Wald[11] and Whitehead and Brunier [15] are usually used inpractice for problem (2) Specifically Sobel and Wald [11]proposed carrying out simultaneous SPRTs of 119867

1versus

1198672and 119867

2versus 119867

3 However when the true parameter

is in the indifference-zones the expected sample size ofthe Sobel-Wald test can be considerably larger than that ofa fixed-sample-size test plan Moreover it is untruncatedsuch that the number of observations required can not bepredetermined an undesirable property in many practicalsituations such as medical trial To reduce the maximumexpected sample size Whitehead and Brunier [15] appliedtwo 2-SPRTs instead of two SPRTs for the component testsat the cost of larger expected sample sizes when the trueparameter does not belong to the indifference-zones

For one-sided composite hypotheses in order to controlthe expected sample sizes Wang et al [20] proposed thedouble sequential weighted probability ratio test (2-SWPRT)based on mixture likelihood ratio statistics and showed thatthe 2-SWPRT is an asymptotically overall optimal test in thesense of asymptoticallyminimizing the expected sample sizeson the indifference-zone Motivated by the attractive proper-ties of the 2-SWPRT we extend the existing work on prob-lem (2) from pointwise optimality to overall performanceoptimality when there are different concerns of interest ondifferent 120579s In particular we propose an optimality criterionto evaluate the overall performance of sequential test planson the indifference-zones for three composite hypotheses andcorrespondingly develop a new sequential test for problem(2) by utilizing two 2-SWPRTs as the component tests toreduce the expected sample sizes We show the proposed testhas a finite stopping time and is asymptotically optimal inthe sense of asymptotically minimizing not only the expectedsample size but also any positive moment of the stoppingtime on the indifference-zones Simulation studies show thatthe proposed test not only has the smallest WESS comparedwith Sobel-Wald and Whitehead-Brunier tests but also issuperior to theWhitehead-Brunier test and comparable withthe Sobel-Wald test when the true parameter does not belongto the indifference-zones Moreover the RMI also shows theproposed test is an efficient method to improve the overallperformance

The rest of this paper is organized as follows In Section 2we review the Sobel-Wald and Whitehead-Brunier testsThe combined double sequential weighted probability ratiotest (denoted by combined 2-SWPRT) is proposed and itsproperties are given in Section 3 Simulation results areprovided in Section 4 and some conclusions are in Section 5All technical details are given in Appendix

2 Methodology Review

For one-sided composite hypotheses1198671versus119867

2 the SPRT

is optimal in the sense that it minimizes the expected samplesizes at 120579

1and 120579

2 and the 2-SPRT has (approximately)

minimal maximum expected sample size over (1205791 1205792) among

all sequential and nonsequential tests with the same errorprobabilities Given the well-known optimality properties ofthe SPRT and 2-SPRT it is natural to use the SPRTs and

2-SPRTs as the component tests to construct the sequentialtests for problem (2) respectively In this section we brieflyreview the Sobel-Wald and Whitehead-Brunier tests

For testing problem (2) the generalization of errors oftypes I and II is expressible in terms of a 3 times 3 error matrix119864 = (120572

119895119896) where 120572

119895119896= 119875[accepting 119867

119895| 119867119896is true] for

119895 119896 = 1 2 3 However under some mild conditions Sobeland Wald [11 pages 504-505] and Armitage [12 pages 142-143] showed that120572

31and12057213are zero which can be verified by

the simulation results in Section 4 It becomes apparent thatin the general case we have at most four ldquodegrees of freedomrdquoin choosing an error matrix Without loss of generality weconsiderΔ = (120591 119889) as a sequential test for problem (2) where120591 is the stopping rule and 119889 is the decision rule (119889 = 119896meansaccepting 119867

119896 119896 = 1 2 3) Set Θ

1= (120579 120579

1] Θ2= [1205792 1205793]

and Θ3= [1205794 120579) Given positive vectors 120572 = (120572

1 1205722) and

120573 = (1205731 1205732) (120572119894+ 120573119894lt 1 119894 = 1 2)

Υ (120572 120573) = Δ = (120591 119889) sup120579isinΘ119894

119875 (119889 = 119894 + 1) le 120572119894

sup120579isinΘ119894+1

119875 (119889 = 119894) le 120573119894 119894 = 1 2

(3)

is the set of all sequential tests with error probabilitiescontrolled by 120572 and 120573

(1) Sobel-Wald Test Since the hypotheses 1198671 1198672 and 119867

3

are ordered the sequential testing of problem (2) can beconstructed by combining the following two one-sided com-posite hypotheses 119878

1and 1198782

1198781 120579 le 120579

1versus 120579

2le 120579 le 120579

3

1198782 1205792le 120579 le 120579

3versus 120579 ge 120579

4

(4)

Sobel and Wald [11] proposed operating 1198781and 119878

2by the

SPRTs simultaneously For all 120579 120588 isin Θ define 119903119899(120579 120588) =

prod119899

119897=1119891(119909119897 120579)119891(119909

119897 120588) The stopping and decision rules of

119878119894(119894 = 1 2) determined by the SRPT are

120591119904

119894= inf 119899 ge 1 119903

119899(1205792119894 1205792119894minus1) notin (119860

119904

119894 119861119904

119894) 119894 = 1 2

119889119904

119894= 119868 (120591

119904

119894lt infin 119903

120591119904

119894

(1205792119894 1205792119894minus1) ge 119861119904

119894) + 119894

(5)

where 119868(sdot) is the indicator function and 119860119904119894and 119861119904

119894(119894 = 1 2)

are the boundary parameters (0 lt 119860119904119894lt 1 lt 119861

119904

119894lt infin) which

are usually set as

119860119904

119894=

120573119894

1 minus 120572119894

119861119904

119894=1 minus 120573119894

120572119894

119894 = 1 2 (6)

to meet requirements on the error probabilities When119861119904

1119861119904

2le 1 and 119860119904

1119860119904

2le 1 Sobel and Wald [11] showed

Mathematical Problems in Engineering 3

the event 1198891199041= 1 119889

119904

2= 3 is impossible The stopping and

decision rules of the Sobel-Wald test are defined as120591119904= max (120591119904

1 120591119904

2)

119889119904=

1 119889119904

1= 1 119889

119904

2= 2

2 119889119904

1= 2 119889

119904

2= 2

3 119889119904

1= 2 119889

119904

2= 3

(7)

The Sobel-Wald test is optimal in the sense that it minimizesthe expected sample sizes at 120579

2119894minus1and 1205792119894(119894 = 1 2) among all

sequential and nonsequential tests whose error probabilitiessatisfy Υ(120572 120573) However its expected sample sizes at otherparameters over Θmay be unsatisfactory

(2) Whitehead-Brunier Test In order to minimize the maxi-mum expected sample size under constraints (3) Whiteheadand Brunier [15] applied the 2-SPRT to operate 119878

1and

1198782 instead of the SPRT As in Lorden [21] let 119870(120579 120588) =

119864120579log[119891(119909 120579)119891(119909 120588)] be the Kullback-Leibler (KL) infor-

mation number Define 120579119894isin (1205792119894minus1 1205792119894) and 119899lowast

119894(119894 = 1 2) by

1003816100381610038161003816log1205721198941003816100381610038161003816

119870 (120579119894 1205792119894minus1)

=

1003816100381610038161003816log1205731198941003816100381610038161003816

119870 (120579119894 1205792119894)

= 119899lowast

119894 119894 = 1 2 (8)

Set 119888lowast119894such that Φ(119888lowast

119894) = minus119886

lowast

2119894(119886lowast

2119894minus1minus 119886lowast

2119894) 119894 = 1 2 where

Φ(sdot) is the cumulative distribution function of the standardnormal distribution 119886lowast

2119894minus1= (120579119894minus1205792119894minus1)119870(120579

119894 1205792119894minus1) and 119886lowast

2119894=

(120579119894minus 1205792119894)119870(120579

119894 1205792119894) 119894 = 1 2 Let

120579lowast

119894= 120579119894+ 119888lowast

119894[119899lowast

11989412059510158401015840(120579119894)]minus12

119894 = 1 2 (9)

The stopping and decision rules of 119878119894(119894 = 1 2) determined by

the 2-SPRT are120591119908

119894= inf 119899 ge 1 119903

119899(120579lowast

119894 1205792119894minus1) ge 119860119908

119894or 119903119899(120579lowast

119894 1205792119894) ge 119861119908

119894

119894 = 1 2

119889119908

119894= 119868 (120591

119908

119894lt infin 119903

120591119908

119894

(120579lowast

119894 1205792119894minus1) ge 119860119908

119894) + 119894

(10)

where 119860119908119894and 119861119908

119894(119894 = 1 2) are the boundary parameters

(0 lt 119860119908119894 119861119908

119894lt infin) The conservative values of 119860119908

119894and 119861119908

119894

are 1120572119894and 1120573

119894 in the sense that the real error probabilities

may be much smaller than 120572119894and 120573

119894(119894 = 1 2) respectively

The stopping and decision rules of the Whitehead-Bruniertest are defined as

120591119908= max (120591119908

1 120591119908

2)

119889119908=

1 119889119908

1= 1 119889

119908

2= 2

2 119889119908

1= 2 119889

119908

2= 2

3 119889119908

1= 2 119889

119908

2= 3

(11)

3 Optimality Criterion andCombined 2-SWPRT

For testing problem (2) if 120579 lt 1205791we prefer to accept 119867

1

and this preference is the stronger the smaller 120579 Similarly

if 120579 gt 1205794we prefer to accept 119867

3 and we prefer to accept

1198672if 1205792lt 120579 lt 120579

3 However we have no strong preference

between 1198671and 119867

2if 120579 isin [120579

1 1205792] and we also have no

strong preference between1198672and119867

3if 120579 isin [120579

3 1205794] In these

cases we need more observations for decision Thus whenthe error probabilities satisfy Υ(120572 120573) we focus on reductionof the expected sample sizes over the indifference-zonesΘ inapplications Let119892(120579) be a nonnegativeweight functionwhichis sectionally continuous on [120579

1 1205792] and [120579

3 1205794] respectively

and satisfies intΘ119892(120579)119889120579 = 1 We define the weighted expected

sample size as

WESS (119892) = intΘ

119864120579120591 sdot 119892 (120579) 119889120579 (12)

to evaluate the overall performance of sequential test plansonΘThe choice of 119892 should be chosen according to practicalneeds (Sobel andWald [11]) For example let 119892(120579) be uniformweights when there are no differences on Θ let 119892(120579) beassigned more weights when we focus more on reducing theexpected sample size on these parameter points As an overallevaluation theWESS(119892) integrates the performances onΘ byweighting the expected sample sizes

Motivated by Wang et al [20] we propose operating1198781and 119878

2by the 2-SWPRT Specifically the stopping and

decision rules of 119878119894(119894 = 1 2) by the 2-SWPRT are

120591lowast

119894= inf 119899 ge 1 119877119894

119899ge 119860119894or 119894119899ge 119861119894) 119894 = 1 2

119889119894= 119868 (120591

lowast

119894lt infin 119877

119894

120591lowast

119894

ge 119860119894) + 119894

(13)

where

119877119894

119899= int

1205792119894

120579lowast

119894

119903119899(120579 1205792119894minus1) 119892 (120579) 119889120579

119894

119899= int

120579lowast

119894

1205792119894minus1

119903119899(120579 1205792119894) 119892 (120579) 119889120579

119894 = 1 2

(14)

where 119860119894and 119861

119894(119894 = 1 2) are the boundary parameters (0 lt

119860119894 119861119894lt infin) Hence the stopping and decision rules of the

combined 2-SWPRT are defined as120591 = max (120591lowast

1 120591lowast

2)

119889 =

1 1198891= 1 119889

2= 2

2 1198891= 2 119889

2= 2

3 1198891= 2 119889

2= 3

(15)

Some features of the combined 2-SWPRT are providedin the following theorems whose proofs are provided inappendices

First we show the error probabilities of the combined2-SWPRT can be easily controlled and the stopping time isfinite

Theorem 1 There exist boundaries 119860119894and 119861

119894(119894 = 1 2) such

that (120591 119889) in (15) belongs to Υ(120572 120573)

4 Mathematical Problems in Engineering

Theorem 2 For any given nonnegative sectionally continuousweight function 119892(120579) the stopping time of the combined 2-SWPRT is finite

Second we prove that the combined 2-SWPRT is asymp-totically optimal on Θ

Definition 3 (120591 119889) isin Υ(120572 120573) is said to be asymptoticallyoptimal on Θ if

lim120572119894+120573119894rarr0

log(120572119894)asymplog(120573119894)

119864120579120591

minus log (120572119894+ 120573119894)= 119869119894(120579) 120579 isin [120579

2119894minus1 1205792119894] (16)

where 119869119894(120579) = min(1119870(120579 120579

2119894minus1) 1119870(120579 120579

2119894)) 119894 = 1 2

Theorem 4 When 119860119894= 120572minus1

119894and 119861

119894= 120573minus1

119894 the (120591 119889) defined

by (15) is asymptotically optimal on Θ

Third we show that any positive moment of the stoppingtime is asymptotically optimal on the indifference-zones

Theorem 5 Under the conditions of Theorem 4 for all 119902 ge 1and 120579 isin [120579

2119894minus1 1205792119894] 119894 = 1 2

lim120572119894+120573119894rarr0

log(120572119894)asymplog(120573119894)

119864120579[(

120591

minus log (120572119894+ 120573119894))

119902

] = (1

119869119894(120579))

119902

119894 = 1 2

(17)

4 Simulation Studies

In this section we conduct simulation studies to examinethe performances of the combined 2-SWPRT the Sobel-Wald test and Whitehead-Brunier test based on the normaland Bernoulli distributions In particular we considered twoweight functions for 119892(120579) as follows (1) uniform weights119892(120579) = 05sum

2

119894=1119868(120579 isin [120579

2119894minus1 1205792119894])(1205792119894minus1205792119894minus1) (2)KLweights

119892(120579) = 05sum2

119894=1119868(120579 isin [120579

2119894minus1 1205792119894])119872119894(120579)[int

1205792119894

1205792119894minus1

119872119894(120579)119889120579]

where119872119894(120579) = max(119870(120579 120579

2119894minus1) 119870(120579 120579

2119894)) As in Wang et al

[20] the corresponding formulations of the statistics 119877119894119899and

119894

119899(119894 = 1 2) can be obtained The boundaries of the tests

are determined through 106 Monte Carlo trials which makethe relative differences between the real error probabilities (1205721015840

119894

and 1205731015840119894) and the required ones (120572

119894and 120573

119894) within 1 that is

|1205721015840

119894minus 120572119894|120572119894lt 1 and |1205731015840

119894minus 120573119894|120573119894lt 1

Given the boundaries we obtained the simulatedWESS(119892) = sum

120579isin119878119864120579120591 sdot 119892(120579) to approximate integral (12) as

follows Let [1205791 1205792] and [120579

3 1205793] be discrete as the finite sets

of parameters 1198781= [1205791 1205791+ Δ 120579

1+ 2Δ 120579

lowast

1 120579

2minus Δ 120579

2]

and 1198782= [1205793 1205793+ Δ 120579

3+ 2Δ 120579

lowast

2 120579

4minus Δ 120579

4] with

increase Δ respectively Denote 119878 = 1198781cup 1198782sub Θ and the

weight function 119892(120579) is calculated based on 120579 isin 119878 thatis 119892(120579) = 05sum

2

119894=1119868(120579 isin 119878

119894)119872119894(120579)[sum

120579isin119878119894119872119894(120579)119889120579] for KL

weights We also compute the RMI to assess the relative

efficiency between different test plans According to Wang etal [20] we define

RMI (119892) = sum120579isin119878

119864120579120591 minus 119878119864

120579120591

119878119864120579120591

sdot 119892 (120579) (18)

where 119878119864120579120591 is the smallest119864

120579120591 among the compared tests that

is the Sobel-Wald test the Whitehead-Brunier test and thecombined 2-SWPRT A test plan with a smaller RMI(119892) valueis considered better in its overall performance

41 Test for the Normal Mean with Known Variance Suppose1198831 1198832 are iid from119873(120579 1) minus120579

1= 1205794= 15 minus120579

2= 1205793=

05 and 1205721= 1205722= 1205731= 1205732= 001 According to Lorden [21]

we have 120579lowast1= minus1 and 120579lowast

2= 1 The stopping boundaries are

obtained as follows(1) for the Sobel-Wald test 119860119904

119894= 0018 and 119861119904

119894= 5573

(2) for the Whitehead-Brunier test 119860119908119894= 119861119908

119894= 3736

(3) for the combined 2-SWPRT 119860119894= 119861119894= 2692 for the

uniform weights and 119860119894= 119861119894= 1378 for the KL

weights respectivelyAs expected we found that 120572

13and 12057231of these three tests are

equal to 0 Set Δ = 005 Through another simulation studywith 105 replications theWESS(119892) andRMI(119892) are presentedin Table 1 Similarly the expected sample sizes for 120579 isin [minus2 2]are illustrated in Figure 1

It is clear that the combined 2-SWPRTs have the smallestWESS(119892) in all cases In fact compared with the Sobel-Waldand Whitehead-Brunier tests the WESS(119892) of the combined2-SWPRT has been reduced by 1136 and 586 for theuniform weights and 813 and 757 for the KL weightsMeanwhile in terms of the RMI(119892) the combined 2-SWPRTalso performs best overall

FromFigure 1 it also can be seen that the expected samplesize of the combined 2-SWPRT is slightly larger than theWhitehead-Brunier test when the true parameter is close to120579lowast

119894(119894 = 1 2) and almost the same as the Sobel-Wald test when

the true parameter is close to 1205792119894minus1

or 1205792119894(119894 = 1 2) When

the true parameter belongs to Θ119896(119896 = 1 2 3) the combined

2-SWPRT performs better than the Whitehead-Brunier testand is comparable with the Sobel-Wald test

42 Test for the True Proportion of a Bernoulli DistributionSuppose 119883

1 1198832 are iid random variables from the

Bernoulli distribution and 119875(1198831= 1) = 119901 = 1 minus 119875(119883

1=

0) (0 lt 119901 lt 1) The three composite hypothesesrsquo testingproblem is

1198670 119901 le 119901

1versus

1198671 1199012le 119901 le 119901

3versus

1198673 119901 ge 119901

4

(19)

where 0 lt 1199011lt 1199012lt 1199013lt 1199014lt 1 Let 120572

1= 1205722= 1205731= 1205732=

001 1199011= 01 119901

2= 03 119901

3= 04 and 119901

4= 07 According to

(9) we have

119901lowast

119894=

log ((1 minus 1199012119894minus1) (1 minus 119901

2119894))

[log (11990121198941199012119894minus1) + log ((1 minus 119901

2119894minus1) (1 minus 119901

2119894))] (20)

Mathematical Problems in Engineering 5

Table 1 WESS(119892) and RMI(119892) for testing normal mean

119892Uniform weights KL weights

Sobel-Wald Whitehead-Brunier Combined 2-SWPRT Sobel-Wald Whitehead-Brunier Combined 2-SWPRTWESS 15231 14342 13511 13818 13734 12694RMI 0268 01652 00183 01629 02117 00178

Table 2 WESS(119892) and RMI(119892) for testing proportion in Bernoulli distribution

119892Uniform weights KL weights

Sobel-Wald Whitehead-Brunier Combined 2-SWPRT Sobel-Wald Whitehead-Brunier Combined 2-SWPRTWESS 48978 44163 43253 43387 41262 40035RMI 02508 00726 00163 01615 00985 00215

24

22

20

18

16

14

12

10

8

6

4

minus20 minus15 minus10 minus05 00 05 10 15 20

120579

Expe

cted

sam

ple s

ize

Sobel-WaldWhitehead-Brunier

Combined 2-SWPRT (uniform)Combined 2-SWPRT (KL)

Figure 1 Expected sample sizes for testing normal mean 1205721= 1205722=

1205731= 1205732= 001 minus120579

1= 1205794= 15 and minus120579

2= 1205793= 05

such that 119901lowast1= 0186 and 119901lowast

2= 0553 in the Whitehead-

Brunier test and combined 2-SWPRT The stopping bound-aries are obtained as follows

(1) for the Sobel-Wald test1198601199041= 0012 119861119904

1= 6652119860119904

2=

0014 and 1198611199042= 7723

(2) for the Whitehead-Brunier test 1198601199081= 3978 119861119908

1=

4651 1198601199082= 4433 and 119861119908

2= 4327

(3) for the combined 2-SWPRT 1198601= 1306 119861

1= 2032

1198602= 1696 and 119861

2= 1627 for the uniform weights

and 1198601= 1962 119861

1= 1435 119860

2= 1718 and 119861

2=

1327 for the KL weights

In this case the values of 12057213= 12057231= 0 Set Δ = 00625

Through another simulation study with 105 replications theWESS(119892) and RMI(119892) are presented in Table 2 Similarly theexpected sample sizes for 119901 isin (0 1) are illustrated in Figure 2

It can be seen from Table 2 that the combined 2-SWPRTstill has the smallest WESS(119892) and RMI(119892) for the Bernoullidistribution Meanwhile from Figure 2 we have similar

80

70

60

50

40

30

20

10

00 01 02 03 04 05 06 07 08 09 10

Expe

cted

sam

ple s

ize

Sobel-WaldWhitehead-Brunier

Combined 2-SWPRT (uniform)Combined 2-SWPRT (KL)

p

Figure 2 Expected sample sizes for testing proportion in Bernoullidistribution 120572

1= 1205722= 1205731= 1205732= 001 119901

2= 01 119901

2= 03 119901

3= 04

and 1199014= 07

conclusions as those in the normal distribution cases inSection 41

5 Summary

In this paper we propose theWESS(119892) to evaluate the overallperformance on the indifference-zones for three compositehypothesesrsquo testing problem In order to minimize WESS(119892)to control the expected sample sizes we developed a newsequential test by utilizing two 2-SWPRTs simultaneouslyWehave shown the proposed test is an asymptotically optimaltest in the sense of asymptotically minimizing the expectedsample sizes on the indifferent-zones

According to the simulation results compared with theSobel-Wald and Whitehead-Brunier tests we conclude thatthe proposed test has the following merits (1) it has thesmallest WESS(119892) and RMI(119892) (2) when the true parameteris close to 120579lowast

119894(119894 = 1 2) the proposed test has comparable

performance with Whitehead-Brunier test when the trueparameter is close to 120579

2119894minus1or 1205792119894(119894 = 1 2) it has almost

6 Mathematical Problems in Engineering

the same results as the Sobel-Wald test when the true param-eter does not belong to Θ the proposed test also performsbetter than the Whitehead-Brunier test and has comparableperformance with the Sobel-Wald test (3) the proposed testis easy to implement and can be extended to multihypothesistesting problems Future work will be concerned with themethod of determining the boundaries in an analytical wayinstead of the Monte Carlo method

Appendix

We provide sketch proofs of Theorems 1 2 4 and 5

Proof ofTheorem 1 LetF119899= 120590(119909

1 119909

119899) 119899 = 1 2 Note

that (1198771119899 119865119899 119899 ge 1) is a supermartingale under 119875

120579 forall120579 isin Θ

1

Therefore for all 120579 isin Θ1

119875120579(119889 = 2) le 119875

120579(1205911lt infin)

le int1205911ltinfin

119860minus1

11198771

1205911119889119875120579

le 119864120579[119860minus1

11198771

1]

(A1)

On the other hand following Lemma 1 of Chen and Hicker-nell [22] for any positive integer119898 and 120579 le 120579

1lt 120582 we have

119864120579[119903119898(120582 1205791)] le 1 (A2)

Thus

119864120579[119860minus1

11198771

1] = 119860

minus1

1int

1205792

120579lowast

1

119864120579[1199031(119905 1205791)] 119892 (119905) 119889119905

le 119860minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905

(A3)

Combining (A1) and (A3) we have

119875120579(119889 = 2) le 119860

minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905 forall120579 isin Θ1 (A4)

In particular setting

1198601= 120572minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905 (A5)

we have sup120579isinΘ1

119875120579(119889 = 2) le 120572

1 Similarly we can prove

that sup120579isinΘ2

119875120579(119889 = 1) le 120573

1 sup120579isinΘ2

119875120579(119889 = 3) le 120572

2 and

sup120579isinΘ3

119875120579(119889 = 2) le 120573

2with 119861

1= 120573minus1

1int120579lowast

1

1205791

119892(119905)119889119905 1198602=

120572minus1

2int1205794

120579lowast

2

119892(119905)119889119905 and 1198612= 120573minus1

2int120579lowast

2

1205793

119892(119905)119889119905 respectively

Proof of Theorem 2 If 119892(120579) is a sectionally continuous func-tion according to Theorem 32 of Wang et al [20] we knowthat (1) for all119860

1gt 0 and 119904

1gt (120595(120579

lowast

1)minus120595(120579

1))(120579lowast

1minus1205791) there

exists 1198791198601(1199041) lt infin such that 1198771

119899ge 1198601when 119899 ge 119879

1198601(1199041) and

119878119899ge 1198991199041 (2) for all 119861

1gt 0 and 119904

1lt (120595(120579

2)minus120595(120579

lowast

1))(1205792minus120579lowast

1)

there exists 1198791198611(1199041) lt infin such that 1

119899ge 1198611when 119899 ge 119879

1198611(1199041)

and 119878119899le 1198991199041 where 119878

119899= sum119899

119897=1119909119897

Noting that120595(120579) is convex we have (120595(120579lowast1)minus120595(120579

1))(120579lowast

1minus

1205791) lt (120595(120579

2) minus 120595(120579

lowast

1))(1205792minus 120579lowast

1) It is easy to choose 119904

1such

that

120595 (120579lowast

1) minus 120595 (120579

1)

120579lowast

1minus 1205791

lt 1199041lt120595 (1205792) minus 120595 (120579

lowast

1)

1205792minus 120579lowast

1

(A6)

Let 11987911986011198611

= max(1198791198601(1199041) 1198791198611(1199041)) Then we have 120591lowast

1le 11987911986011198611

Similarly for all 119860

2gt 0 and 119861

2gt 0 we can prove that there

exist 1198791198602(1199042) lt infin and 119879

1198612(1199042) lt infin such that 120591lowast

2le 11987911986021198612

Thus we have

120591 le max (11987911986011198611

11987911986021198612

) (A7)

Proof of Theorem 4 Using Hoeffding inequality (see Hoeffd-ing [23]) we know

lim inf1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)ge 1198691(120579) 120579 isin [120579

1 1205792] (A8)

so it suffices to show

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)le 1198691(120579) 120579 isin [120579

1 1205792] (A9)

According to Theorem 33 of Wang et al [20] for all 120579 isin

[120579lowast

1 1205792]

lim1205721rarr0

1205911

log (1198601)=

1

119870 (120579 1205791)

(as 119875120579) (A10)

Since 1198601= 120572minus1

1 when 120572

1+ 1205731rarr 0 and log(120572

1) asymp log(120573

1)

we have

minus log (1205721+ 1205731) 997888rarr minus log (120572

1) = log (119860

1) (A11)

Therefore for all 120579 isin [120579lowast1 1205792]

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591lowast

1

minus log (1205721+ 1205731)le lim1205721rarr0

1198641205791205911

log (1198601)=

1

119870 (120579 1205791)

(A12)

Similarly for all 120579 isin [1205791 120579lowast

1] we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591lowast

1

minus log (1205721+ 1205731)le lim1205731rarr0

1198641205791205911

log (1198611)=

1

119870 (120579 1205792)

(A13)

Combining two inequalities (A12) and (A13) we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)le 1198691(120579) 120579 isin [120579

1 1205792] (A14)

According to (A8) and (A14)

lim1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)= 1198691(120579) 120579 isin [120579

1 1205792] (A15)

Mathematical Problems in Engineering 7

Similarly we can prove that

lim1205722+1205732rarr0

log(1205722)asymplog(1205732)

119864120579120591

minus log (1205722+ 1205732)= 1198692(120579) 120579 isin [120579

3 1205794] (A16)

Proof of Theorem 5 Using Lemma 36 of Chen [24] for all119902 ge 1 we know

lim1205721rarr0

119864120579[(

1205911

minus log (1205721))

119902

] =1

[119870 (120579 1205791)]119902 (120579

lowast

1le 120579 le 120579

2)

lim1205731rarr0

119864120579[(

1205911

minus log (1205731))

119902

] =1

[119870 (120579 1205792)]119902 (120579

1le 120579 le 120579

lowast

1)

(A17)

Similar to Theorem 4 for all 120579 isin [120579lowast1 1205792] we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591lowast

1

minus log (1205721+ 1205731))]

119902

le lim1205721rarr0

119864120579[(

1205911

minus log (1205721))]

119902

=1

[119870 (120579 1205791)]119902

(A18)

For all 120579 isin [1205791 120579lowast

1] there is

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591lowast

1

minus log (1205721+ 1205731))]

119902

le lim1205731rarr0

119864120579[(

1205911

minus log (1205731))]

119902

=1

[119870 (120579 1205792)]119902

(A19)

According to (A18) (A19) and Hoeffding inequality wehave

lim1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591

minus log (1205721+ 1205731))

119902

]

= (1

1198691(120579))

119902

120579 isin [1205791 1205792]

(A20)

Similarly we can prove that

lim1205722+1205732rarr0

log(1205722)asymplog(1205732)

119864120579[(

120591

minus log (1205722+ 1205732))

119902

]

= (1

1198691(120579))

119902

120579 isin [1205793 1205794]

(A21)

Conflict of Interests

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

Acknowledgments

The authors would like to thank the Academic Edi-tor Antonino Laudani and an anonymous referee fortheir insightful comments and suggestions on this paperwhich have led to significant improvements This workwas supported by the Postdoctoral Science Foundation ofChina (2014M560317) the National Science Fund of China(11271135 11471119 11371142 and 11101156) the FundamentalResearch Funds for the Central Universities the 111 Project(B14019) and the Programof Shanghai Subject Chief Scientist(14XD1401600)

References

[1] J J Goeman A Solari and T Stijnen ldquoThree-sided hypothesistesting simultaneous testing of superiority equivalence andinferiorityrdquo Statistics in Medicine vol 29 no 20 pp 2117ndash21252010

[2] K S Fu Sequential Methods in Pattern Recognition and Learn-ing Academic Press New York NY USA 1968

[3] T McMillen and P Holmes ldquoThe dynamics of choice amongmultiple alternativesrdquo Journal of Mathematical Psychology vol50 no 1 pp 30ndash57 2006

[4] J J Bussgang ldquoSequential methods in radar detectionrdquo Proceed-ings of the IEEE vol 58 no 5 pp 731ndash743 1970

[5] S L Anderson ldquoSimple method of comparing the breakingstrength of two yarnsrdquo Journal of the Textle Institute vol 45 pp472ndash479 1954

[6] Y Li X L Pu and F Tsung ldquoAdaptive charting schemesbased on double sequential probability ratio testsrdquo Quality andReliability Engineering International vol 25 no 1 pp 21ndash392009

[7] I V Pavlov ldquoA sequential procedure for testingmany compositehypothesesrdquoTheory of Probability amp Its Applications vol 32 no1 pp 138ndash142 1988

[8] C W Baum and V V Veeravalli ldquoA sequential procedurefor multihypothesis testingrdquo IEEE Transactions on InformationTheory vol 40 no 6 pp 1994ndash2007 1994

[9] V P Dragalin A G Tartakovsky and V V Veeravalli ldquoMul-tihypothesis sequential probability ratio tests I asymptoticoptimalityrdquo IEEE Transactions on Information Theory vol 45no 7 pp 2448ndash2461 1999

[10] V P Dragalin A G Tartakovsky and V V Veeravalli ldquoMul-tihypothesis sequential probability ratio tests II Accurateasymptotic expansions for the expected sample sizerdquo IEEETransactions on Information Theory vol 46 no 4 pp 1366ndash1383 2000

[11] M Sobel and A Wald ldquoA sequential decision procedure forchoosing one of three hypotheses concerning the unknownmean of a normal distributionrdquo Annals of Mathematical Statis-tics vol 20 pp 502ndash522 1949

[12] P Armitage ldquoSequential analysis withmore than two alternativehypotheses and its relation to discriminant function analysisrdquoJournal of the Royal Statistical Society Series B Methodologicalvol 12 pp 137ndash144 1950

[13] G Simons ldquoLower bounds for average sample number ofsequential multihypothesis testsrdquo Annals of MathematicalStatistics vol 38 pp 1343ndash1364 1967

8 Mathematical Problems in Engineering

[14] G Lorden ldquoLikelihood ratio tests for sequential k-decisionproblemsrdquo Annals of Mathematical Statistics vol 43 pp 1412ndash1427 1972

[15] J Whitehead and H Brunier ldquoThe double triangular test asequential test for the two-sided alternative with early stoppingunder the null hypothesisrdquo Sequential Analysis Design Methodsamp Applications vol 9 no 2 pp 117ndash136 1990

[16] Y Li and X Pu ldquoHypothesis designs for three-hypothesis testproblemsrdquo Mathematical Problems in Engineering vol 2010Article ID 393095 15 pages 2010

[17] Y Li and X L Pu ldquoA method for designing three-hypothesistest problems and sequential schemesrdquo Communications inStatisticsmdashSimulation and Computation vol 39 no 9 pp 1690ndash1708 2010

[18] V P Dragalin and A A Novikov ldquoAdaptive sequential testsfor composite hypothesesrdquo Surveys of Applied and IndustrialMathematics vol 6 pp 387ndash398 1999

[19] T L Lai ldquoSequential multiple hypothesis testing and efficientfault detection-isolation in stochastic systemsrdquo IEEE Transac-tions on Information Theory vol 46 no 2 pp 595ndash608 2000

[20] L Wang D D Xiang X L Pu and Y Li ldquoA double sequen-tial weighted probability ratio test for one-sided compositehypothesesrdquo Communications in StatisticsTheory andMethodsvol 42 no 20 pp 3678ndash3695 2013

[21] G Lorden ldquo2-SPRTrsquos and the modified Kiefer-Weiss problemof minimizing an expected sample sizerdquoTheAnnals of Statisticsvol 4 no 2 pp 281ndash291 1976

[22] J D Chen and F J Hickernell ldquoA class of asymptotically optimalsequential tests for composite hypothesesrdquo Science in ChinaSeries A vol 37 no 11 pp 1314ndash1324 1994

[23] W Hoeffding ldquoLower bounds for the expected sample sizeand the average risk of a sequential procedurerdquo Annals ofMathematical Statistics vol 31 pp 352ndash368 1960

[24] J D Chen ldquoAsymptotic optimality for one class of truncatedsequential testsrdquo Science in China Series A MathematicsPhysics Astronomy vol 30 pp 30ndash41 1991

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

the event 1198891199041= 1 119889

119904

2= 3 is impossible The stopping and

decision rules of the Sobel-Wald test are defined as120591119904= max (120591119904

1 120591119904

2)

119889119904=

1 119889119904

1= 1 119889

119904

2= 2

2 119889119904

1= 2 119889

119904

2= 2

3 119889119904

1= 2 119889

119904

2= 3

(7)

The Sobel-Wald test is optimal in the sense that it minimizesthe expected sample sizes at 120579

2119894minus1and 1205792119894(119894 = 1 2) among all

sequential and nonsequential tests whose error probabilitiessatisfy Υ(120572 120573) However its expected sample sizes at otherparameters over Θmay be unsatisfactory

(2) Whitehead-Brunier Test In order to minimize the maxi-mum expected sample size under constraints (3) Whiteheadand Brunier [15] applied the 2-SPRT to operate 119878

1and

1198782 instead of the SPRT As in Lorden [21] let 119870(120579 120588) =

119864120579log[119891(119909 120579)119891(119909 120588)] be the Kullback-Leibler (KL) infor-

mation number Define 120579119894isin (1205792119894minus1 1205792119894) and 119899lowast

119894(119894 = 1 2) by

1003816100381610038161003816log1205721198941003816100381610038161003816

119870 (120579119894 1205792119894minus1)

=

1003816100381610038161003816log1205731198941003816100381610038161003816

119870 (120579119894 1205792119894)

= 119899lowast

119894 119894 = 1 2 (8)

Set 119888lowast119894such that Φ(119888lowast

119894) = minus119886

lowast

2119894(119886lowast

2119894minus1minus 119886lowast

2119894) 119894 = 1 2 where

Φ(sdot) is the cumulative distribution function of the standardnormal distribution 119886lowast

2119894minus1= (120579119894minus1205792119894minus1)119870(120579

119894 1205792119894minus1) and 119886lowast

2119894=

(120579119894minus 1205792119894)119870(120579

119894 1205792119894) 119894 = 1 2 Let

120579lowast

119894= 120579119894+ 119888lowast

119894[119899lowast

11989412059510158401015840(120579119894)]minus12

119894 = 1 2 (9)

The stopping and decision rules of 119878119894(119894 = 1 2) determined by

the 2-SPRT are120591119908

119894= inf 119899 ge 1 119903

119899(120579lowast

119894 1205792119894minus1) ge 119860119908

119894or 119903119899(120579lowast

119894 1205792119894) ge 119861119908

119894

119894 = 1 2

119889119908

119894= 119868 (120591

119908

119894lt infin 119903

120591119908

119894

(120579lowast

119894 1205792119894minus1) ge 119860119908

119894) + 119894

(10)

where 119860119908119894and 119861119908

119894(119894 = 1 2) are the boundary parameters

(0 lt 119860119908119894 119861119908

119894lt infin) The conservative values of 119860119908

119894and 119861119908

119894

are 1120572119894and 1120573

119894 in the sense that the real error probabilities

may be much smaller than 120572119894and 120573

119894(119894 = 1 2) respectively

The stopping and decision rules of the Whitehead-Bruniertest are defined as

120591119908= max (120591119908

1 120591119908

2)

119889119908=

1 119889119908

1= 1 119889

119908

2= 2

2 119889119908

1= 2 119889

119908

2= 2

3 119889119908

1= 2 119889

119908

2= 3

(11)

3 Optimality Criterion andCombined 2-SWPRT

For testing problem (2) if 120579 lt 1205791we prefer to accept 119867

1

and this preference is the stronger the smaller 120579 Similarly

if 120579 gt 1205794we prefer to accept 119867

3 and we prefer to accept

1198672if 1205792lt 120579 lt 120579

3 However we have no strong preference

between 1198671and 119867

2if 120579 isin [120579

1 1205792] and we also have no

strong preference between1198672and119867

3if 120579 isin [120579

3 1205794] In these

cases we need more observations for decision Thus whenthe error probabilities satisfy Υ(120572 120573) we focus on reductionof the expected sample sizes over the indifference-zonesΘ inapplications Let119892(120579) be a nonnegativeweight functionwhichis sectionally continuous on [120579

1 1205792] and [120579

3 1205794] respectively

and satisfies intΘ119892(120579)119889120579 = 1 We define the weighted expected

sample size as

WESS (119892) = intΘ

119864120579120591 sdot 119892 (120579) 119889120579 (12)

to evaluate the overall performance of sequential test plansonΘThe choice of 119892 should be chosen according to practicalneeds (Sobel andWald [11]) For example let 119892(120579) be uniformweights when there are no differences on Θ let 119892(120579) beassigned more weights when we focus more on reducing theexpected sample size on these parameter points As an overallevaluation theWESS(119892) integrates the performances onΘ byweighting the expected sample sizes

Motivated by Wang et al [20] we propose operating1198781and 119878

2by the 2-SWPRT Specifically the stopping and

decision rules of 119878119894(119894 = 1 2) by the 2-SWPRT are

120591lowast

119894= inf 119899 ge 1 119877119894

119899ge 119860119894or 119894119899ge 119861119894) 119894 = 1 2

119889119894= 119868 (120591

lowast

119894lt infin 119877

119894

120591lowast

119894

ge 119860119894) + 119894

(13)

where

119877119894

119899= int

1205792119894

120579lowast

119894

119903119899(120579 1205792119894minus1) 119892 (120579) 119889120579

119894

119899= int

120579lowast

119894

1205792119894minus1

119903119899(120579 1205792119894) 119892 (120579) 119889120579

119894 = 1 2

(14)

where 119860119894and 119861

119894(119894 = 1 2) are the boundary parameters (0 lt

119860119894 119861119894lt infin) Hence the stopping and decision rules of the

combined 2-SWPRT are defined as120591 = max (120591lowast

1 120591lowast

2)

119889 =

1 1198891= 1 119889

2= 2

2 1198891= 2 119889

2= 2

3 1198891= 2 119889

2= 3

(15)

Some features of the combined 2-SWPRT are providedin the following theorems whose proofs are provided inappendices

First we show the error probabilities of the combined2-SWPRT can be easily controlled and the stopping time isfinite

Theorem 1 There exist boundaries 119860119894and 119861

119894(119894 = 1 2) such

that (120591 119889) in (15) belongs to Υ(120572 120573)

4 Mathematical Problems in Engineering

Theorem 2 For any given nonnegative sectionally continuousweight function 119892(120579) the stopping time of the combined 2-SWPRT is finite

Second we prove that the combined 2-SWPRT is asymp-totically optimal on Θ

Definition 3 (120591 119889) isin Υ(120572 120573) is said to be asymptoticallyoptimal on Θ if

lim120572119894+120573119894rarr0

log(120572119894)asymplog(120573119894)

119864120579120591

minus log (120572119894+ 120573119894)= 119869119894(120579) 120579 isin [120579

2119894minus1 1205792119894] (16)

where 119869119894(120579) = min(1119870(120579 120579

2119894minus1) 1119870(120579 120579

2119894)) 119894 = 1 2

Theorem 4 When 119860119894= 120572minus1

119894and 119861

119894= 120573minus1

119894 the (120591 119889) defined

by (15) is asymptotically optimal on Θ

Third we show that any positive moment of the stoppingtime is asymptotically optimal on the indifference-zones

Theorem 5 Under the conditions of Theorem 4 for all 119902 ge 1and 120579 isin [120579

2119894minus1 1205792119894] 119894 = 1 2

lim120572119894+120573119894rarr0

log(120572119894)asymplog(120573119894)

119864120579[(

120591

minus log (120572119894+ 120573119894))

119902

] = (1

119869119894(120579))

119902

119894 = 1 2

(17)

4 Simulation Studies

In this section we conduct simulation studies to examinethe performances of the combined 2-SWPRT the Sobel-Wald test and Whitehead-Brunier test based on the normaland Bernoulli distributions In particular we considered twoweight functions for 119892(120579) as follows (1) uniform weights119892(120579) = 05sum

2

119894=1119868(120579 isin [120579

2119894minus1 1205792119894])(1205792119894minus1205792119894minus1) (2)KLweights

119892(120579) = 05sum2

119894=1119868(120579 isin [120579

2119894minus1 1205792119894])119872119894(120579)[int

1205792119894

1205792119894minus1

119872119894(120579)119889120579]

where119872119894(120579) = max(119870(120579 120579

2119894minus1) 119870(120579 120579

2119894)) As in Wang et al

[20] the corresponding formulations of the statistics 119877119894119899and

119894

119899(119894 = 1 2) can be obtained The boundaries of the tests

are determined through 106 Monte Carlo trials which makethe relative differences between the real error probabilities (1205721015840

119894

and 1205731015840119894) and the required ones (120572

119894and 120573

119894) within 1 that is

|1205721015840

119894minus 120572119894|120572119894lt 1 and |1205731015840

119894minus 120573119894|120573119894lt 1

Given the boundaries we obtained the simulatedWESS(119892) = sum

120579isin119878119864120579120591 sdot 119892(120579) to approximate integral (12) as

follows Let [1205791 1205792] and [120579

3 1205793] be discrete as the finite sets

of parameters 1198781= [1205791 1205791+ Δ 120579

1+ 2Δ 120579

lowast

1 120579

2minus Δ 120579

2]

and 1198782= [1205793 1205793+ Δ 120579

3+ 2Δ 120579

lowast

2 120579

4minus Δ 120579

4] with

increase Δ respectively Denote 119878 = 1198781cup 1198782sub Θ and the

weight function 119892(120579) is calculated based on 120579 isin 119878 thatis 119892(120579) = 05sum

2

119894=1119868(120579 isin 119878

119894)119872119894(120579)[sum

120579isin119878119894119872119894(120579)119889120579] for KL

weights We also compute the RMI to assess the relative

efficiency between different test plans According to Wang etal [20] we define

RMI (119892) = sum120579isin119878

119864120579120591 minus 119878119864

120579120591

119878119864120579120591

sdot 119892 (120579) (18)

where 119878119864120579120591 is the smallest119864

120579120591 among the compared tests that

is the Sobel-Wald test the Whitehead-Brunier test and thecombined 2-SWPRT A test plan with a smaller RMI(119892) valueis considered better in its overall performance

41 Test for the Normal Mean with Known Variance Suppose1198831 1198832 are iid from119873(120579 1) minus120579

1= 1205794= 15 minus120579

2= 1205793=

05 and 1205721= 1205722= 1205731= 1205732= 001 According to Lorden [21]

we have 120579lowast1= minus1 and 120579lowast

2= 1 The stopping boundaries are

obtained as follows(1) for the Sobel-Wald test 119860119904

119894= 0018 and 119861119904

119894= 5573

(2) for the Whitehead-Brunier test 119860119908119894= 119861119908

119894= 3736

(3) for the combined 2-SWPRT 119860119894= 119861119894= 2692 for the

uniform weights and 119860119894= 119861119894= 1378 for the KL

weights respectivelyAs expected we found that 120572

13and 12057231of these three tests are

equal to 0 Set Δ = 005 Through another simulation studywith 105 replications theWESS(119892) andRMI(119892) are presentedin Table 1 Similarly the expected sample sizes for 120579 isin [minus2 2]are illustrated in Figure 1

It is clear that the combined 2-SWPRTs have the smallestWESS(119892) in all cases In fact compared with the Sobel-Waldand Whitehead-Brunier tests the WESS(119892) of the combined2-SWPRT has been reduced by 1136 and 586 for theuniform weights and 813 and 757 for the KL weightsMeanwhile in terms of the RMI(119892) the combined 2-SWPRTalso performs best overall

FromFigure 1 it also can be seen that the expected samplesize of the combined 2-SWPRT is slightly larger than theWhitehead-Brunier test when the true parameter is close to120579lowast

119894(119894 = 1 2) and almost the same as the Sobel-Wald test when

the true parameter is close to 1205792119894minus1

or 1205792119894(119894 = 1 2) When

the true parameter belongs to Θ119896(119896 = 1 2 3) the combined

2-SWPRT performs better than the Whitehead-Brunier testand is comparable with the Sobel-Wald test

42 Test for the True Proportion of a Bernoulli DistributionSuppose 119883

1 1198832 are iid random variables from the

Bernoulli distribution and 119875(1198831= 1) = 119901 = 1 minus 119875(119883

1=

0) (0 lt 119901 lt 1) The three composite hypothesesrsquo testingproblem is

1198670 119901 le 119901

1versus

1198671 1199012le 119901 le 119901

3versus

1198673 119901 ge 119901

4

(19)

where 0 lt 1199011lt 1199012lt 1199013lt 1199014lt 1 Let 120572

1= 1205722= 1205731= 1205732=

001 1199011= 01 119901

2= 03 119901

3= 04 and 119901

4= 07 According to

(9) we have

119901lowast

119894=

log ((1 minus 1199012119894minus1) (1 minus 119901

2119894))

[log (11990121198941199012119894minus1) + log ((1 minus 119901

2119894minus1) (1 minus 119901

2119894))] (20)

Mathematical Problems in Engineering 5

Table 1 WESS(119892) and RMI(119892) for testing normal mean

119892Uniform weights KL weights

Sobel-Wald Whitehead-Brunier Combined 2-SWPRT Sobel-Wald Whitehead-Brunier Combined 2-SWPRTWESS 15231 14342 13511 13818 13734 12694RMI 0268 01652 00183 01629 02117 00178

Table 2 WESS(119892) and RMI(119892) for testing proportion in Bernoulli distribution

119892Uniform weights KL weights

Sobel-Wald Whitehead-Brunier Combined 2-SWPRT Sobel-Wald Whitehead-Brunier Combined 2-SWPRTWESS 48978 44163 43253 43387 41262 40035RMI 02508 00726 00163 01615 00985 00215

24

22

20

18

16

14

12

10

8

6

4

minus20 minus15 minus10 minus05 00 05 10 15 20

120579

Expe

cted

sam

ple s

ize

Sobel-WaldWhitehead-Brunier

Combined 2-SWPRT (uniform)Combined 2-SWPRT (KL)

Figure 1 Expected sample sizes for testing normal mean 1205721= 1205722=

1205731= 1205732= 001 minus120579

1= 1205794= 15 and minus120579

2= 1205793= 05

such that 119901lowast1= 0186 and 119901lowast

2= 0553 in the Whitehead-

Brunier test and combined 2-SWPRT The stopping bound-aries are obtained as follows

(1) for the Sobel-Wald test1198601199041= 0012 119861119904

1= 6652119860119904

2=

0014 and 1198611199042= 7723

(2) for the Whitehead-Brunier test 1198601199081= 3978 119861119908

1=

4651 1198601199082= 4433 and 119861119908

2= 4327

(3) for the combined 2-SWPRT 1198601= 1306 119861

1= 2032

1198602= 1696 and 119861

2= 1627 for the uniform weights

and 1198601= 1962 119861

1= 1435 119860

2= 1718 and 119861

2=

1327 for the KL weights

In this case the values of 12057213= 12057231= 0 Set Δ = 00625

Through another simulation study with 105 replications theWESS(119892) and RMI(119892) are presented in Table 2 Similarly theexpected sample sizes for 119901 isin (0 1) are illustrated in Figure 2

It can be seen from Table 2 that the combined 2-SWPRTstill has the smallest WESS(119892) and RMI(119892) for the Bernoullidistribution Meanwhile from Figure 2 we have similar

80

70

60

50

40

30

20

10

00 01 02 03 04 05 06 07 08 09 10

Expe

cted

sam

ple s

ize

Sobel-WaldWhitehead-Brunier

Combined 2-SWPRT (uniform)Combined 2-SWPRT (KL)

p

Figure 2 Expected sample sizes for testing proportion in Bernoullidistribution 120572

1= 1205722= 1205731= 1205732= 001 119901

2= 01 119901

2= 03 119901

3= 04

and 1199014= 07

conclusions as those in the normal distribution cases inSection 41

5 Summary

In this paper we propose theWESS(119892) to evaluate the overallperformance on the indifference-zones for three compositehypothesesrsquo testing problem In order to minimize WESS(119892)to control the expected sample sizes we developed a newsequential test by utilizing two 2-SWPRTs simultaneouslyWehave shown the proposed test is an asymptotically optimaltest in the sense of asymptotically minimizing the expectedsample sizes on the indifferent-zones

According to the simulation results compared with theSobel-Wald and Whitehead-Brunier tests we conclude thatthe proposed test has the following merits (1) it has thesmallest WESS(119892) and RMI(119892) (2) when the true parameteris close to 120579lowast

119894(119894 = 1 2) the proposed test has comparable

performance with Whitehead-Brunier test when the trueparameter is close to 120579

2119894minus1or 1205792119894(119894 = 1 2) it has almost

6 Mathematical Problems in Engineering

the same results as the Sobel-Wald test when the true param-eter does not belong to Θ the proposed test also performsbetter than the Whitehead-Brunier test and has comparableperformance with the Sobel-Wald test (3) the proposed testis easy to implement and can be extended to multihypothesistesting problems Future work will be concerned with themethod of determining the boundaries in an analytical wayinstead of the Monte Carlo method

Appendix

We provide sketch proofs of Theorems 1 2 4 and 5

Proof ofTheorem 1 LetF119899= 120590(119909

1 119909

119899) 119899 = 1 2 Note

that (1198771119899 119865119899 119899 ge 1) is a supermartingale under 119875

120579 forall120579 isin Θ

1

Therefore for all 120579 isin Θ1

119875120579(119889 = 2) le 119875

120579(1205911lt infin)

le int1205911ltinfin

119860minus1

11198771

1205911119889119875120579

le 119864120579[119860minus1

11198771

1]

(A1)

On the other hand following Lemma 1 of Chen and Hicker-nell [22] for any positive integer119898 and 120579 le 120579

1lt 120582 we have

119864120579[119903119898(120582 1205791)] le 1 (A2)

Thus

119864120579[119860minus1

11198771

1] = 119860

minus1

1int

1205792

120579lowast

1

119864120579[1199031(119905 1205791)] 119892 (119905) 119889119905

le 119860minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905

(A3)

Combining (A1) and (A3) we have

119875120579(119889 = 2) le 119860

minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905 forall120579 isin Θ1 (A4)

In particular setting

1198601= 120572minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905 (A5)

we have sup120579isinΘ1

119875120579(119889 = 2) le 120572

1 Similarly we can prove

that sup120579isinΘ2

119875120579(119889 = 1) le 120573

1 sup120579isinΘ2

119875120579(119889 = 3) le 120572

2 and

sup120579isinΘ3

119875120579(119889 = 2) le 120573

2with 119861

1= 120573minus1

1int120579lowast

1

1205791

119892(119905)119889119905 1198602=

120572minus1

2int1205794

120579lowast

2

119892(119905)119889119905 and 1198612= 120573minus1

2int120579lowast

2

1205793

119892(119905)119889119905 respectively

Proof of Theorem 2 If 119892(120579) is a sectionally continuous func-tion according to Theorem 32 of Wang et al [20] we knowthat (1) for all119860

1gt 0 and 119904

1gt (120595(120579

lowast

1)minus120595(120579

1))(120579lowast

1minus1205791) there

exists 1198791198601(1199041) lt infin such that 1198771

119899ge 1198601when 119899 ge 119879

1198601(1199041) and

119878119899ge 1198991199041 (2) for all 119861

1gt 0 and 119904

1lt (120595(120579

2)minus120595(120579

lowast

1))(1205792minus120579lowast

1)

there exists 1198791198611(1199041) lt infin such that 1

119899ge 1198611when 119899 ge 119879

1198611(1199041)

and 119878119899le 1198991199041 where 119878

119899= sum119899

119897=1119909119897

Noting that120595(120579) is convex we have (120595(120579lowast1)minus120595(120579

1))(120579lowast

1minus

1205791) lt (120595(120579

2) minus 120595(120579

lowast

1))(1205792minus 120579lowast

1) It is easy to choose 119904

1such

that

120595 (120579lowast

1) minus 120595 (120579

1)

120579lowast

1minus 1205791

lt 1199041lt120595 (1205792) minus 120595 (120579

lowast

1)

1205792minus 120579lowast

1

(A6)

Let 11987911986011198611

= max(1198791198601(1199041) 1198791198611(1199041)) Then we have 120591lowast

1le 11987911986011198611

Similarly for all 119860

2gt 0 and 119861

2gt 0 we can prove that there

exist 1198791198602(1199042) lt infin and 119879

1198612(1199042) lt infin such that 120591lowast

2le 11987911986021198612

Thus we have

120591 le max (11987911986011198611

11987911986021198612

) (A7)

Proof of Theorem 4 Using Hoeffding inequality (see Hoeffd-ing [23]) we know

lim inf1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)ge 1198691(120579) 120579 isin [120579

1 1205792] (A8)

so it suffices to show

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)le 1198691(120579) 120579 isin [120579

1 1205792] (A9)

According to Theorem 33 of Wang et al [20] for all 120579 isin

[120579lowast

1 1205792]

lim1205721rarr0

1205911

log (1198601)=

1

119870 (120579 1205791)

(as 119875120579) (A10)

Since 1198601= 120572minus1

1 when 120572

1+ 1205731rarr 0 and log(120572

1) asymp log(120573

1)

we have

minus log (1205721+ 1205731) 997888rarr minus log (120572

1) = log (119860

1) (A11)

Therefore for all 120579 isin [120579lowast1 1205792]

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591lowast

1

minus log (1205721+ 1205731)le lim1205721rarr0

1198641205791205911

log (1198601)=

1

119870 (120579 1205791)

(A12)

Similarly for all 120579 isin [1205791 120579lowast

1] we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591lowast

1

minus log (1205721+ 1205731)le lim1205731rarr0

1198641205791205911

log (1198611)=

1

119870 (120579 1205792)

(A13)

Combining two inequalities (A12) and (A13) we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)le 1198691(120579) 120579 isin [120579

1 1205792] (A14)

According to (A8) and (A14)

lim1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)= 1198691(120579) 120579 isin [120579

1 1205792] (A15)

Mathematical Problems in Engineering 7

Similarly we can prove that

lim1205722+1205732rarr0

log(1205722)asymplog(1205732)

119864120579120591

minus log (1205722+ 1205732)= 1198692(120579) 120579 isin [120579

3 1205794] (A16)

Proof of Theorem 5 Using Lemma 36 of Chen [24] for all119902 ge 1 we know

lim1205721rarr0

119864120579[(

1205911

minus log (1205721))

119902

] =1

[119870 (120579 1205791)]119902 (120579

lowast

1le 120579 le 120579

2)

lim1205731rarr0

119864120579[(

1205911

minus log (1205731))

119902

] =1

[119870 (120579 1205792)]119902 (120579

1le 120579 le 120579

lowast

1)

(A17)

Similar to Theorem 4 for all 120579 isin [120579lowast1 1205792] we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591lowast

1

minus log (1205721+ 1205731))]

119902

le lim1205721rarr0

119864120579[(

1205911

minus log (1205721))]

119902

=1

[119870 (120579 1205791)]119902

(A18)

For all 120579 isin [1205791 120579lowast

1] there is

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591lowast

1

minus log (1205721+ 1205731))]

119902

le lim1205731rarr0

119864120579[(

1205911

minus log (1205731))]

119902

=1

[119870 (120579 1205792)]119902

(A19)

According to (A18) (A19) and Hoeffding inequality wehave

lim1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591

minus log (1205721+ 1205731))

119902

]

= (1

1198691(120579))

119902

120579 isin [1205791 1205792]

(A20)

Similarly we can prove that

lim1205722+1205732rarr0

log(1205722)asymplog(1205732)

119864120579[(

120591

minus log (1205722+ 1205732))

119902

]

= (1

1198691(120579))

119902

120579 isin [1205793 1205794]

(A21)

Conflict of Interests

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

Acknowledgments

The authors would like to thank the Academic Edi-tor Antonino Laudani and an anonymous referee fortheir insightful comments and suggestions on this paperwhich have led to significant improvements This workwas supported by the Postdoctoral Science Foundation ofChina (2014M560317) the National Science Fund of China(11271135 11471119 11371142 and 11101156) the FundamentalResearch Funds for the Central Universities the 111 Project(B14019) and the Programof Shanghai Subject Chief Scientist(14XD1401600)

References

[1] J J Goeman A Solari and T Stijnen ldquoThree-sided hypothesistesting simultaneous testing of superiority equivalence andinferiorityrdquo Statistics in Medicine vol 29 no 20 pp 2117ndash21252010

[2] K S Fu Sequential Methods in Pattern Recognition and Learn-ing Academic Press New York NY USA 1968

[3] T McMillen and P Holmes ldquoThe dynamics of choice amongmultiple alternativesrdquo Journal of Mathematical Psychology vol50 no 1 pp 30ndash57 2006

[4] J J Bussgang ldquoSequential methods in radar detectionrdquo Proceed-ings of the IEEE vol 58 no 5 pp 731ndash743 1970

[5] S L Anderson ldquoSimple method of comparing the breakingstrength of two yarnsrdquo Journal of the Textle Institute vol 45 pp472ndash479 1954

[6] Y Li X L Pu and F Tsung ldquoAdaptive charting schemesbased on double sequential probability ratio testsrdquo Quality andReliability Engineering International vol 25 no 1 pp 21ndash392009

[7] I V Pavlov ldquoA sequential procedure for testingmany compositehypothesesrdquoTheory of Probability amp Its Applications vol 32 no1 pp 138ndash142 1988

[8] C W Baum and V V Veeravalli ldquoA sequential procedurefor multihypothesis testingrdquo IEEE Transactions on InformationTheory vol 40 no 6 pp 1994ndash2007 1994

[9] V P Dragalin A G Tartakovsky and V V Veeravalli ldquoMul-tihypothesis sequential probability ratio tests I asymptoticoptimalityrdquo IEEE Transactions on Information Theory vol 45no 7 pp 2448ndash2461 1999

[10] V P Dragalin A G Tartakovsky and V V Veeravalli ldquoMul-tihypothesis sequential probability ratio tests II Accurateasymptotic expansions for the expected sample sizerdquo IEEETransactions on Information Theory vol 46 no 4 pp 1366ndash1383 2000

[11] M Sobel and A Wald ldquoA sequential decision procedure forchoosing one of three hypotheses concerning the unknownmean of a normal distributionrdquo Annals of Mathematical Statis-tics vol 20 pp 502ndash522 1949

[12] P Armitage ldquoSequential analysis withmore than two alternativehypotheses and its relation to discriminant function analysisrdquoJournal of the Royal Statistical Society Series B Methodologicalvol 12 pp 137ndash144 1950

[13] G Simons ldquoLower bounds for average sample number ofsequential multihypothesis testsrdquo Annals of MathematicalStatistics vol 38 pp 1343ndash1364 1967

8 Mathematical Problems in Engineering

[14] G Lorden ldquoLikelihood ratio tests for sequential k-decisionproblemsrdquo Annals of Mathematical Statistics vol 43 pp 1412ndash1427 1972

[15] J Whitehead and H Brunier ldquoThe double triangular test asequential test for the two-sided alternative with early stoppingunder the null hypothesisrdquo Sequential Analysis Design Methodsamp Applications vol 9 no 2 pp 117ndash136 1990

[16] Y Li and X Pu ldquoHypothesis designs for three-hypothesis testproblemsrdquo Mathematical Problems in Engineering vol 2010Article ID 393095 15 pages 2010

[17] Y Li and X L Pu ldquoA method for designing three-hypothesistest problems and sequential schemesrdquo Communications inStatisticsmdashSimulation and Computation vol 39 no 9 pp 1690ndash1708 2010

[18] V P Dragalin and A A Novikov ldquoAdaptive sequential testsfor composite hypothesesrdquo Surveys of Applied and IndustrialMathematics vol 6 pp 387ndash398 1999

[19] T L Lai ldquoSequential multiple hypothesis testing and efficientfault detection-isolation in stochastic systemsrdquo IEEE Transac-tions on Information Theory vol 46 no 2 pp 595ndash608 2000

[20] L Wang D D Xiang X L Pu and Y Li ldquoA double sequen-tial weighted probability ratio test for one-sided compositehypothesesrdquo Communications in StatisticsTheory andMethodsvol 42 no 20 pp 3678ndash3695 2013

[21] G Lorden ldquo2-SPRTrsquos and the modified Kiefer-Weiss problemof minimizing an expected sample sizerdquoTheAnnals of Statisticsvol 4 no 2 pp 281ndash291 1976

[22] J D Chen and F J Hickernell ldquoA class of asymptotically optimalsequential tests for composite hypothesesrdquo Science in ChinaSeries A vol 37 no 11 pp 1314ndash1324 1994

[23] W Hoeffding ldquoLower bounds for the expected sample sizeand the average risk of a sequential procedurerdquo Annals ofMathematical Statistics vol 31 pp 352ndash368 1960

[24] J D Chen ldquoAsymptotic optimality for one class of truncatedsequential testsrdquo Science in China Series A MathematicsPhysics Astronomy vol 30 pp 30ndash41 1991

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

4 Mathematical Problems in Engineering

Theorem 2 For any given nonnegative sectionally continuousweight function 119892(120579) the stopping time of the combined 2-SWPRT is finite

Second we prove that the combined 2-SWPRT is asymp-totically optimal on Θ

Definition 3 (120591 119889) isin Υ(120572 120573) is said to be asymptoticallyoptimal on Θ if

lim120572119894+120573119894rarr0

log(120572119894)asymplog(120573119894)

119864120579120591

minus log (120572119894+ 120573119894)= 119869119894(120579) 120579 isin [120579

2119894minus1 1205792119894] (16)

where 119869119894(120579) = min(1119870(120579 120579

2119894minus1) 1119870(120579 120579

2119894)) 119894 = 1 2

Theorem 4 When 119860119894= 120572minus1

119894and 119861

119894= 120573minus1

119894 the (120591 119889) defined

by (15) is asymptotically optimal on Θ

Third we show that any positive moment of the stoppingtime is asymptotically optimal on the indifference-zones

Theorem 5 Under the conditions of Theorem 4 for all 119902 ge 1and 120579 isin [120579

2119894minus1 1205792119894] 119894 = 1 2

lim120572119894+120573119894rarr0

log(120572119894)asymplog(120573119894)

119864120579[(

120591

minus log (120572119894+ 120573119894))

119902

] = (1

119869119894(120579))

119902

119894 = 1 2

(17)

4 Simulation Studies

In this section we conduct simulation studies to examinethe performances of the combined 2-SWPRT the Sobel-Wald test and Whitehead-Brunier test based on the normaland Bernoulli distributions In particular we considered twoweight functions for 119892(120579) as follows (1) uniform weights119892(120579) = 05sum

2

119894=1119868(120579 isin [120579

2119894minus1 1205792119894])(1205792119894minus1205792119894minus1) (2)KLweights

119892(120579) = 05sum2

119894=1119868(120579 isin [120579

2119894minus1 1205792119894])119872119894(120579)[int

1205792119894

1205792119894minus1

119872119894(120579)119889120579]

where119872119894(120579) = max(119870(120579 120579

2119894minus1) 119870(120579 120579

2119894)) As in Wang et al

[20] the corresponding formulations of the statistics 119877119894119899and

119894

119899(119894 = 1 2) can be obtained The boundaries of the tests

are determined through 106 Monte Carlo trials which makethe relative differences between the real error probabilities (1205721015840

119894

and 1205731015840119894) and the required ones (120572

119894and 120573

119894) within 1 that is

|1205721015840

119894minus 120572119894|120572119894lt 1 and |1205731015840

119894minus 120573119894|120573119894lt 1

Given the boundaries we obtained the simulatedWESS(119892) = sum

120579isin119878119864120579120591 sdot 119892(120579) to approximate integral (12) as

follows Let [1205791 1205792] and [120579

3 1205793] be discrete as the finite sets

of parameters 1198781= [1205791 1205791+ Δ 120579

1+ 2Δ 120579

lowast

1 120579

2minus Δ 120579

2]

and 1198782= [1205793 1205793+ Δ 120579

3+ 2Δ 120579

lowast

2 120579

4minus Δ 120579

4] with

increase Δ respectively Denote 119878 = 1198781cup 1198782sub Θ and the

weight function 119892(120579) is calculated based on 120579 isin 119878 thatis 119892(120579) = 05sum

2

119894=1119868(120579 isin 119878

119894)119872119894(120579)[sum

120579isin119878119894119872119894(120579)119889120579] for KL

weights We also compute the RMI to assess the relative

efficiency between different test plans According to Wang etal [20] we define

RMI (119892) = sum120579isin119878

119864120579120591 minus 119878119864

120579120591

119878119864120579120591

sdot 119892 (120579) (18)

where 119878119864120579120591 is the smallest119864

120579120591 among the compared tests that

is the Sobel-Wald test the Whitehead-Brunier test and thecombined 2-SWPRT A test plan with a smaller RMI(119892) valueis considered better in its overall performance

41 Test for the Normal Mean with Known Variance Suppose1198831 1198832 are iid from119873(120579 1) minus120579

1= 1205794= 15 minus120579

2= 1205793=

05 and 1205721= 1205722= 1205731= 1205732= 001 According to Lorden [21]

we have 120579lowast1= minus1 and 120579lowast

2= 1 The stopping boundaries are

obtained as follows(1) for the Sobel-Wald test 119860119904

119894= 0018 and 119861119904

119894= 5573

(2) for the Whitehead-Brunier test 119860119908119894= 119861119908

119894= 3736

(3) for the combined 2-SWPRT 119860119894= 119861119894= 2692 for the

uniform weights and 119860119894= 119861119894= 1378 for the KL

weights respectivelyAs expected we found that 120572

13and 12057231of these three tests are

equal to 0 Set Δ = 005 Through another simulation studywith 105 replications theWESS(119892) andRMI(119892) are presentedin Table 1 Similarly the expected sample sizes for 120579 isin [minus2 2]are illustrated in Figure 1

It is clear that the combined 2-SWPRTs have the smallestWESS(119892) in all cases In fact compared with the Sobel-Waldand Whitehead-Brunier tests the WESS(119892) of the combined2-SWPRT has been reduced by 1136 and 586 for theuniform weights and 813 and 757 for the KL weightsMeanwhile in terms of the RMI(119892) the combined 2-SWPRTalso performs best overall

FromFigure 1 it also can be seen that the expected samplesize of the combined 2-SWPRT is slightly larger than theWhitehead-Brunier test when the true parameter is close to120579lowast

119894(119894 = 1 2) and almost the same as the Sobel-Wald test when

the true parameter is close to 1205792119894minus1

or 1205792119894(119894 = 1 2) When

the true parameter belongs to Θ119896(119896 = 1 2 3) the combined

2-SWPRT performs better than the Whitehead-Brunier testand is comparable with the Sobel-Wald test

42 Test for the True Proportion of a Bernoulli DistributionSuppose 119883

1 1198832 are iid random variables from the

Bernoulli distribution and 119875(1198831= 1) = 119901 = 1 minus 119875(119883

1=

0) (0 lt 119901 lt 1) The three composite hypothesesrsquo testingproblem is

1198670 119901 le 119901

1versus

1198671 1199012le 119901 le 119901

3versus

1198673 119901 ge 119901

4

(19)

where 0 lt 1199011lt 1199012lt 1199013lt 1199014lt 1 Let 120572

1= 1205722= 1205731= 1205732=

001 1199011= 01 119901

2= 03 119901

3= 04 and 119901

4= 07 According to

(9) we have

119901lowast

119894=

log ((1 minus 1199012119894minus1) (1 minus 119901

2119894))

[log (11990121198941199012119894minus1) + log ((1 minus 119901

2119894minus1) (1 minus 119901

2119894))] (20)

Mathematical Problems in Engineering 5

Table 1 WESS(119892) and RMI(119892) for testing normal mean

119892Uniform weights KL weights

Sobel-Wald Whitehead-Brunier Combined 2-SWPRT Sobel-Wald Whitehead-Brunier Combined 2-SWPRTWESS 15231 14342 13511 13818 13734 12694RMI 0268 01652 00183 01629 02117 00178

Table 2 WESS(119892) and RMI(119892) for testing proportion in Bernoulli distribution

119892Uniform weights KL weights

Sobel-Wald Whitehead-Brunier Combined 2-SWPRT Sobel-Wald Whitehead-Brunier Combined 2-SWPRTWESS 48978 44163 43253 43387 41262 40035RMI 02508 00726 00163 01615 00985 00215

24

22

20

18

16

14

12

10

8

6

4

minus20 minus15 minus10 minus05 00 05 10 15 20

120579

Expe

cted

sam

ple s

ize

Sobel-WaldWhitehead-Brunier

Combined 2-SWPRT (uniform)Combined 2-SWPRT (KL)

Figure 1 Expected sample sizes for testing normal mean 1205721= 1205722=

1205731= 1205732= 001 minus120579

1= 1205794= 15 and minus120579

2= 1205793= 05

such that 119901lowast1= 0186 and 119901lowast

2= 0553 in the Whitehead-

Brunier test and combined 2-SWPRT The stopping bound-aries are obtained as follows

(1) for the Sobel-Wald test1198601199041= 0012 119861119904

1= 6652119860119904

2=

0014 and 1198611199042= 7723

(2) for the Whitehead-Brunier test 1198601199081= 3978 119861119908

1=

4651 1198601199082= 4433 and 119861119908

2= 4327

(3) for the combined 2-SWPRT 1198601= 1306 119861

1= 2032

1198602= 1696 and 119861

2= 1627 for the uniform weights

and 1198601= 1962 119861

1= 1435 119860

2= 1718 and 119861

2=

1327 for the KL weights

In this case the values of 12057213= 12057231= 0 Set Δ = 00625

Through another simulation study with 105 replications theWESS(119892) and RMI(119892) are presented in Table 2 Similarly theexpected sample sizes for 119901 isin (0 1) are illustrated in Figure 2

It can be seen from Table 2 that the combined 2-SWPRTstill has the smallest WESS(119892) and RMI(119892) for the Bernoullidistribution Meanwhile from Figure 2 we have similar

80

70

60

50

40

30

20

10

00 01 02 03 04 05 06 07 08 09 10

Expe

cted

sam

ple s

ize

Sobel-WaldWhitehead-Brunier

Combined 2-SWPRT (uniform)Combined 2-SWPRT (KL)

p

Figure 2 Expected sample sizes for testing proportion in Bernoullidistribution 120572

1= 1205722= 1205731= 1205732= 001 119901

2= 01 119901

2= 03 119901

3= 04

and 1199014= 07

conclusions as those in the normal distribution cases inSection 41

5 Summary

In this paper we propose theWESS(119892) to evaluate the overallperformance on the indifference-zones for three compositehypothesesrsquo testing problem In order to minimize WESS(119892)to control the expected sample sizes we developed a newsequential test by utilizing two 2-SWPRTs simultaneouslyWehave shown the proposed test is an asymptotically optimaltest in the sense of asymptotically minimizing the expectedsample sizes on the indifferent-zones

According to the simulation results compared with theSobel-Wald and Whitehead-Brunier tests we conclude thatthe proposed test has the following merits (1) it has thesmallest WESS(119892) and RMI(119892) (2) when the true parameteris close to 120579lowast

119894(119894 = 1 2) the proposed test has comparable

performance with Whitehead-Brunier test when the trueparameter is close to 120579

2119894minus1or 1205792119894(119894 = 1 2) it has almost

6 Mathematical Problems in Engineering

the same results as the Sobel-Wald test when the true param-eter does not belong to Θ the proposed test also performsbetter than the Whitehead-Brunier test and has comparableperformance with the Sobel-Wald test (3) the proposed testis easy to implement and can be extended to multihypothesistesting problems Future work will be concerned with themethod of determining the boundaries in an analytical wayinstead of the Monte Carlo method

Appendix

We provide sketch proofs of Theorems 1 2 4 and 5

Proof ofTheorem 1 LetF119899= 120590(119909

1 119909

119899) 119899 = 1 2 Note

that (1198771119899 119865119899 119899 ge 1) is a supermartingale under 119875

120579 forall120579 isin Θ

1

Therefore for all 120579 isin Θ1

119875120579(119889 = 2) le 119875

120579(1205911lt infin)

le int1205911ltinfin

119860minus1

11198771

1205911119889119875120579

le 119864120579[119860minus1

11198771

1]

(A1)

On the other hand following Lemma 1 of Chen and Hicker-nell [22] for any positive integer119898 and 120579 le 120579

1lt 120582 we have

119864120579[119903119898(120582 1205791)] le 1 (A2)

Thus

119864120579[119860minus1

11198771

1] = 119860

minus1

1int

1205792

120579lowast

1

119864120579[1199031(119905 1205791)] 119892 (119905) 119889119905

le 119860minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905

(A3)

Combining (A1) and (A3) we have

119875120579(119889 = 2) le 119860

minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905 forall120579 isin Θ1 (A4)

In particular setting

1198601= 120572minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905 (A5)

we have sup120579isinΘ1

119875120579(119889 = 2) le 120572

1 Similarly we can prove

that sup120579isinΘ2

119875120579(119889 = 1) le 120573

1 sup120579isinΘ2

119875120579(119889 = 3) le 120572

2 and

sup120579isinΘ3

119875120579(119889 = 2) le 120573

2with 119861

1= 120573minus1

1int120579lowast

1

1205791

119892(119905)119889119905 1198602=

120572minus1

2int1205794

120579lowast

2

119892(119905)119889119905 and 1198612= 120573minus1

2int120579lowast

2

1205793

119892(119905)119889119905 respectively

Proof of Theorem 2 If 119892(120579) is a sectionally continuous func-tion according to Theorem 32 of Wang et al [20] we knowthat (1) for all119860

1gt 0 and 119904

1gt (120595(120579

lowast

1)minus120595(120579

1))(120579lowast

1minus1205791) there

exists 1198791198601(1199041) lt infin such that 1198771

119899ge 1198601when 119899 ge 119879

1198601(1199041) and

119878119899ge 1198991199041 (2) for all 119861

1gt 0 and 119904

1lt (120595(120579

2)minus120595(120579

lowast

1))(1205792minus120579lowast

1)

there exists 1198791198611(1199041) lt infin such that 1

119899ge 1198611when 119899 ge 119879

1198611(1199041)

and 119878119899le 1198991199041 where 119878

119899= sum119899

119897=1119909119897

Noting that120595(120579) is convex we have (120595(120579lowast1)minus120595(120579

1))(120579lowast

1minus

1205791) lt (120595(120579

2) minus 120595(120579

lowast

1))(1205792minus 120579lowast

1) It is easy to choose 119904

1such

that

120595 (120579lowast

1) minus 120595 (120579

1)

120579lowast

1minus 1205791

lt 1199041lt120595 (1205792) minus 120595 (120579

lowast

1)

1205792minus 120579lowast

1

(A6)

Let 11987911986011198611

= max(1198791198601(1199041) 1198791198611(1199041)) Then we have 120591lowast

1le 11987911986011198611

Similarly for all 119860

2gt 0 and 119861

2gt 0 we can prove that there

exist 1198791198602(1199042) lt infin and 119879

1198612(1199042) lt infin such that 120591lowast

2le 11987911986021198612

Thus we have

120591 le max (11987911986011198611

11987911986021198612

) (A7)

Proof of Theorem 4 Using Hoeffding inequality (see Hoeffd-ing [23]) we know

lim inf1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)ge 1198691(120579) 120579 isin [120579

1 1205792] (A8)

so it suffices to show

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)le 1198691(120579) 120579 isin [120579

1 1205792] (A9)

According to Theorem 33 of Wang et al [20] for all 120579 isin

[120579lowast

1 1205792]

lim1205721rarr0

1205911

log (1198601)=

1

119870 (120579 1205791)

(as 119875120579) (A10)

Since 1198601= 120572minus1

1 when 120572

1+ 1205731rarr 0 and log(120572

1) asymp log(120573

1)

we have

minus log (1205721+ 1205731) 997888rarr minus log (120572

1) = log (119860

1) (A11)

Therefore for all 120579 isin [120579lowast1 1205792]

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591lowast

1

minus log (1205721+ 1205731)le lim1205721rarr0

1198641205791205911

log (1198601)=

1

119870 (120579 1205791)

(A12)

Similarly for all 120579 isin [1205791 120579lowast

1] we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591lowast

1

minus log (1205721+ 1205731)le lim1205731rarr0

1198641205791205911

log (1198611)=

1

119870 (120579 1205792)

(A13)

Combining two inequalities (A12) and (A13) we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)le 1198691(120579) 120579 isin [120579

1 1205792] (A14)

According to (A8) and (A14)

lim1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)= 1198691(120579) 120579 isin [120579

1 1205792] (A15)

Mathematical Problems in Engineering 7

Similarly we can prove that

lim1205722+1205732rarr0

log(1205722)asymplog(1205732)

119864120579120591

minus log (1205722+ 1205732)= 1198692(120579) 120579 isin [120579

3 1205794] (A16)

Proof of Theorem 5 Using Lemma 36 of Chen [24] for all119902 ge 1 we know

lim1205721rarr0

119864120579[(

1205911

minus log (1205721))

119902

] =1

[119870 (120579 1205791)]119902 (120579

lowast

1le 120579 le 120579

2)

lim1205731rarr0

119864120579[(

1205911

minus log (1205731))

119902

] =1

[119870 (120579 1205792)]119902 (120579

1le 120579 le 120579

lowast

1)

(A17)

Similar to Theorem 4 for all 120579 isin [120579lowast1 1205792] we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591lowast

1

minus log (1205721+ 1205731))]

119902

le lim1205721rarr0

119864120579[(

1205911

minus log (1205721))]

119902

=1

[119870 (120579 1205791)]119902

(A18)

For all 120579 isin [1205791 120579lowast

1] there is

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591lowast

1

minus log (1205721+ 1205731))]

119902

le lim1205731rarr0

119864120579[(

1205911

minus log (1205731))]

119902

=1

[119870 (120579 1205792)]119902

(A19)

According to (A18) (A19) and Hoeffding inequality wehave

lim1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591

minus log (1205721+ 1205731))

119902

]

= (1

1198691(120579))

119902

120579 isin [1205791 1205792]

(A20)

Similarly we can prove that

lim1205722+1205732rarr0

log(1205722)asymplog(1205732)

119864120579[(

120591

minus log (1205722+ 1205732))

119902

]

= (1

1198691(120579))

119902

120579 isin [1205793 1205794]

(A21)

Conflict of Interests

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

Acknowledgments

The authors would like to thank the Academic Edi-tor Antonino Laudani and an anonymous referee fortheir insightful comments and suggestions on this paperwhich have led to significant improvements This workwas supported by the Postdoctoral Science Foundation ofChina (2014M560317) the National Science Fund of China(11271135 11471119 11371142 and 11101156) the FundamentalResearch Funds for the Central Universities the 111 Project(B14019) and the Programof Shanghai Subject Chief Scientist(14XD1401600)

References

[1] J J Goeman A Solari and T Stijnen ldquoThree-sided hypothesistesting simultaneous testing of superiority equivalence andinferiorityrdquo Statistics in Medicine vol 29 no 20 pp 2117ndash21252010

[2] K S Fu Sequential Methods in Pattern Recognition and Learn-ing Academic Press New York NY USA 1968

[3] T McMillen and P Holmes ldquoThe dynamics of choice amongmultiple alternativesrdquo Journal of Mathematical Psychology vol50 no 1 pp 30ndash57 2006

[4] J J Bussgang ldquoSequential methods in radar detectionrdquo Proceed-ings of the IEEE vol 58 no 5 pp 731ndash743 1970

[5] S L Anderson ldquoSimple method of comparing the breakingstrength of two yarnsrdquo Journal of the Textle Institute vol 45 pp472ndash479 1954

[6] Y Li X L Pu and F Tsung ldquoAdaptive charting schemesbased on double sequential probability ratio testsrdquo Quality andReliability Engineering International vol 25 no 1 pp 21ndash392009

[7] I V Pavlov ldquoA sequential procedure for testingmany compositehypothesesrdquoTheory of Probability amp Its Applications vol 32 no1 pp 138ndash142 1988

[8] C W Baum and V V Veeravalli ldquoA sequential procedurefor multihypothesis testingrdquo IEEE Transactions on InformationTheory vol 40 no 6 pp 1994ndash2007 1994

[9] V P Dragalin A G Tartakovsky and V V Veeravalli ldquoMul-tihypothesis sequential probability ratio tests I asymptoticoptimalityrdquo IEEE Transactions on Information Theory vol 45no 7 pp 2448ndash2461 1999

[10] V P Dragalin A G Tartakovsky and V V Veeravalli ldquoMul-tihypothesis sequential probability ratio tests II Accurateasymptotic expansions for the expected sample sizerdquo IEEETransactions on Information Theory vol 46 no 4 pp 1366ndash1383 2000

[11] M Sobel and A Wald ldquoA sequential decision procedure forchoosing one of three hypotheses concerning the unknownmean of a normal distributionrdquo Annals of Mathematical Statis-tics vol 20 pp 502ndash522 1949

[12] P Armitage ldquoSequential analysis withmore than two alternativehypotheses and its relation to discriminant function analysisrdquoJournal of the Royal Statistical Society Series B Methodologicalvol 12 pp 137ndash144 1950

[13] G Simons ldquoLower bounds for average sample number ofsequential multihypothesis testsrdquo Annals of MathematicalStatistics vol 38 pp 1343ndash1364 1967

8 Mathematical Problems in Engineering

[14] G Lorden ldquoLikelihood ratio tests for sequential k-decisionproblemsrdquo Annals of Mathematical Statistics vol 43 pp 1412ndash1427 1972

[15] J Whitehead and H Brunier ldquoThe double triangular test asequential test for the two-sided alternative with early stoppingunder the null hypothesisrdquo Sequential Analysis Design Methodsamp Applications vol 9 no 2 pp 117ndash136 1990

[16] Y Li and X Pu ldquoHypothesis designs for three-hypothesis testproblemsrdquo Mathematical Problems in Engineering vol 2010Article ID 393095 15 pages 2010

[17] Y Li and X L Pu ldquoA method for designing three-hypothesistest problems and sequential schemesrdquo Communications inStatisticsmdashSimulation and Computation vol 39 no 9 pp 1690ndash1708 2010

[18] V P Dragalin and A A Novikov ldquoAdaptive sequential testsfor composite hypothesesrdquo Surveys of Applied and IndustrialMathematics vol 6 pp 387ndash398 1999

[19] T L Lai ldquoSequential multiple hypothesis testing and efficientfault detection-isolation in stochastic systemsrdquo IEEE Transac-tions on Information Theory vol 46 no 2 pp 595ndash608 2000

[20] L Wang D D Xiang X L Pu and Y Li ldquoA double sequen-tial weighted probability ratio test for one-sided compositehypothesesrdquo Communications in StatisticsTheory andMethodsvol 42 no 20 pp 3678ndash3695 2013

[21] G Lorden ldquo2-SPRTrsquos and the modified Kiefer-Weiss problemof minimizing an expected sample sizerdquoTheAnnals of Statisticsvol 4 no 2 pp 281ndash291 1976

[22] J D Chen and F J Hickernell ldquoA class of asymptotically optimalsequential tests for composite hypothesesrdquo Science in ChinaSeries A vol 37 no 11 pp 1314ndash1324 1994

[23] W Hoeffding ldquoLower bounds for the expected sample sizeand the average risk of a sequential procedurerdquo Annals ofMathematical Statistics vol 31 pp 352ndash368 1960

[24] J D Chen ldquoAsymptotic optimality for one class of truncatedsequential testsrdquo Science in China Series A MathematicsPhysics Astronomy vol 30 pp 30ndash41 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Table 1 WESS(119892) and RMI(119892) for testing normal mean

119892Uniform weights KL weights

Sobel-Wald Whitehead-Brunier Combined 2-SWPRT Sobel-Wald Whitehead-Brunier Combined 2-SWPRTWESS 15231 14342 13511 13818 13734 12694RMI 0268 01652 00183 01629 02117 00178

Table 2 WESS(119892) and RMI(119892) for testing proportion in Bernoulli distribution

119892Uniform weights KL weights

Sobel-Wald Whitehead-Brunier Combined 2-SWPRT Sobel-Wald Whitehead-Brunier Combined 2-SWPRTWESS 48978 44163 43253 43387 41262 40035RMI 02508 00726 00163 01615 00985 00215

24

22

20

18

16

14

12

10

8

6

4

minus20 minus15 minus10 minus05 00 05 10 15 20

120579

Expe

cted

sam

ple s

ize

Sobel-WaldWhitehead-Brunier

Combined 2-SWPRT (uniform)Combined 2-SWPRT (KL)

Figure 1 Expected sample sizes for testing normal mean 1205721= 1205722=

1205731= 1205732= 001 minus120579

1= 1205794= 15 and minus120579

2= 1205793= 05

such that 119901lowast1= 0186 and 119901lowast

2= 0553 in the Whitehead-

Brunier test and combined 2-SWPRT The stopping bound-aries are obtained as follows

(1) for the Sobel-Wald test1198601199041= 0012 119861119904

1= 6652119860119904

2=

0014 and 1198611199042= 7723

(2) for the Whitehead-Brunier test 1198601199081= 3978 119861119908

1=

4651 1198601199082= 4433 and 119861119908

2= 4327

(3) for the combined 2-SWPRT 1198601= 1306 119861

1= 2032

1198602= 1696 and 119861

2= 1627 for the uniform weights

and 1198601= 1962 119861

1= 1435 119860

2= 1718 and 119861

2=

1327 for the KL weights

In this case the values of 12057213= 12057231= 0 Set Δ = 00625

Through another simulation study with 105 replications theWESS(119892) and RMI(119892) are presented in Table 2 Similarly theexpected sample sizes for 119901 isin (0 1) are illustrated in Figure 2

It can be seen from Table 2 that the combined 2-SWPRTstill has the smallest WESS(119892) and RMI(119892) for the Bernoullidistribution Meanwhile from Figure 2 we have similar

80

70

60

50

40

30

20

10

00 01 02 03 04 05 06 07 08 09 10

Expe

cted

sam

ple s

ize

Sobel-WaldWhitehead-Brunier

Combined 2-SWPRT (uniform)Combined 2-SWPRT (KL)

p

Figure 2 Expected sample sizes for testing proportion in Bernoullidistribution 120572

1= 1205722= 1205731= 1205732= 001 119901

2= 01 119901

2= 03 119901

3= 04

and 1199014= 07

conclusions as those in the normal distribution cases inSection 41

5 Summary

In this paper we propose theWESS(119892) to evaluate the overallperformance on the indifference-zones for three compositehypothesesrsquo testing problem In order to minimize WESS(119892)to control the expected sample sizes we developed a newsequential test by utilizing two 2-SWPRTs simultaneouslyWehave shown the proposed test is an asymptotically optimaltest in the sense of asymptotically minimizing the expectedsample sizes on the indifferent-zones

According to the simulation results compared with theSobel-Wald and Whitehead-Brunier tests we conclude thatthe proposed test has the following merits (1) it has thesmallest WESS(119892) and RMI(119892) (2) when the true parameteris close to 120579lowast

119894(119894 = 1 2) the proposed test has comparable

performance with Whitehead-Brunier test when the trueparameter is close to 120579

2119894minus1or 1205792119894(119894 = 1 2) it has almost

6 Mathematical Problems in Engineering

the same results as the Sobel-Wald test when the true param-eter does not belong to Θ the proposed test also performsbetter than the Whitehead-Brunier test and has comparableperformance with the Sobel-Wald test (3) the proposed testis easy to implement and can be extended to multihypothesistesting problems Future work will be concerned with themethod of determining the boundaries in an analytical wayinstead of the Monte Carlo method

Appendix

We provide sketch proofs of Theorems 1 2 4 and 5

Proof ofTheorem 1 LetF119899= 120590(119909

1 119909

119899) 119899 = 1 2 Note

that (1198771119899 119865119899 119899 ge 1) is a supermartingale under 119875

120579 forall120579 isin Θ

1

Therefore for all 120579 isin Θ1

119875120579(119889 = 2) le 119875

120579(1205911lt infin)

le int1205911ltinfin

119860minus1

11198771

1205911119889119875120579

le 119864120579[119860minus1

11198771

1]

(A1)

On the other hand following Lemma 1 of Chen and Hicker-nell [22] for any positive integer119898 and 120579 le 120579

1lt 120582 we have

119864120579[119903119898(120582 1205791)] le 1 (A2)

Thus

119864120579[119860minus1

11198771

1] = 119860

minus1

1int

1205792

120579lowast

1

119864120579[1199031(119905 1205791)] 119892 (119905) 119889119905

le 119860minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905

(A3)

Combining (A1) and (A3) we have

119875120579(119889 = 2) le 119860

minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905 forall120579 isin Θ1 (A4)

In particular setting

1198601= 120572minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905 (A5)

we have sup120579isinΘ1

119875120579(119889 = 2) le 120572

1 Similarly we can prove

that sup120579isinΘ2

119875120579(119889 = 1) le 120573

1 sup120579isinΘ2

119875120579(119889 = 3) le 120572

2 and

sup120579isinΘ3

119875120579(119889 = 2) le 120573

2with 119861

1= 120573minus1

1int120579lowast

1

1205791

119892(119905)119889119905 1198602=

120572minus1

2int1205794

120579lowast

2

119892(119905)119889119905 and 1198612= 120573minus1

2int120579lowast

2

1205793

119892(119905)119889119905 respectively

Proof of Theorem 2 If 119892(120579) is a sectionally continuous func-tion according to Theorem 32 of Wang et al [20] we knowthat (1) for all119860

1gt 0 and 119904

1gt (120595(120579

lowast

1)minus120595(120579

1))(120579lowast

1minus1205791) there

exists 1198791198601(1199041) lt infin such that 1198771

119899ge 1198601when 119899 ge 119879

1198601(1199041) and

119878119899ge 1198991199041 (2) for all 119861

1gt 0 and 119904

1lt (120595(120579

2)minus120595(120579

lowast

1))(1205792minus120579lowast

1)

there exists 1198791198611(1199041) lt infin such that 1

119899ge 1198611when 119899 ge 119879

1198611(1199041)

and 119878119899le 1198991199041 where 119878

119899= sum119899

119897=1119909119897

Noting that120595(120579) is convex we have (120595(120579lowast1)minus120595(120579

1))(120579lowast

1minus

1205791) lt (120595(120579

2) minus 120595(120579

lowast

1))(1205792minus 120579lowast

1) It is easy to choose 119904

1such

that

120595 (120579lowast

1) minus 120595 (120579

1)

120579lowast

1minus 1205791

lt 1199041lt120595 (1205792) minus 120595 (120579

lowast

1)

1205792minus 120579lowast

1

(A6)

Let 11987911986011198611

= max(1198791198601(1199041) 1198791198611(1199041)) Then we have 120591lowast

1le 11987911986011198611

Similarly for all 119860

2gt 0 and 119861

2gt 0 we can prove that there

exist 1198791198602(1199042) lt infin and 119879

1198612(1199042) lt infin such that 120591lowast

2le 11987911986021198612

Thus we have

120591 le max (11987911986011198611

11987911986021198612

) (A7)

Proof of Theorem 4 Using Hoeffding inequality (see Hoeffd-ing [23]) we know

lim inf1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)ge 1198691(120579) 120579 isin [120579

1 1205792] (A8)

so it suffices to show

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)le 1198691(120579) 120579 isin [120579

1 1205792] (A9)

According to Theorem 33 of Wang et al [20] for all 120579 isin

[120579lowast

1 1205792]

lim1205721rarr0

1205911

log (1198601)=

1

119870 (120579 1205791)

(as 119875120579) (A10)

Since 1198601= 120572minus1

1 when 120572

1+ 1205731rarr 0 and log(120572

1) asymp log(120573

1)

we have

minus log (1205721+ 1205731) 997888rarr minus log (120572

1) = log (119860

1) (A11)

Therefore for all 120579 isin [120579lowast1 1205792]

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591lowast

1

minus log (1205721+ 1205731)le lim1205721rarr0

1198641205791205911

log (1198601)=

1

119870 (120579 1205791)

(A12)

Similarly for all 120579 isin [1205791 120579lowast

1] we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591lowast

1

minus log (1205721+ 1205731)le lim1205731rarr0

1198641205791205911

log (1198611)=

1

119870 (120579 1205792)

(A13)

Combining two inequalities (A12) and (A13) we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)le 1198691(120579) 120579 isin [120579

1 1205792] (A14)

According to (A8) and (A14)

lim1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)= 1198691(120579) 120579 isin [120579

1 1205792] (A15)

Mathematical Problems in Engineering 7

Similarly we can prove that

lim1205722+1205732rarr0

log(1205722)asymplog(1205732)

119864120579120591

minus log (1205722+ 1205732)= 1198692(120579) 120579 isin [120579

3 1205794] (A16)

Proof of Theorem 5 Using Lemma 36 of Chen [24] for all119902 ge 1 we know

lim1205721rarr0

119864120579[(

1205911

minus log (1205721))

119902

] =1

[119870 (120579 1205791)]119902 (120579

lowast

1le 120579 le 120579

2)

lim1205731rarr0

119864120579[(

1205911

minus log (1205731))

119902

] =1

[119870 (120579 1205792)]119902 (120579

1le 120579 le 120579

lowast

1)

(A17)

Similar to Theorem 4 for all 120579 isin [120579lowast1 1205792] we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591lowast

1

minus log (1205721+ 1205731))]

119902

le lim1205721rarr0

119864120579[(

1205911

minus log (1205721))]

119902

=1

[119870 (120579 1205791)]119902

(A18)

For all 120579 isin [1205791 120579lowast

1] there is

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591lowast

1

minus log (1205721+ 1205731))]

119902

le lim1205731rarr0

119864120579[(

1205911

minus log (1205731))]

119902

=1

[119870 (120579 1205792)]119902

(A19)

According to (A18) (A19) and Hoeffding inequality wehave

lim1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591

minus log (1205721+ 1205731))

119902

]

= (1

1198691(120579))

119902

120579 isin [1205791 1205792]

(A20)

Similarly we can prove that

lim1205722+1205732rarr0

log(1205722)asymplog(1205732)

119864120579[(

120591

minus log (1205722+ 1205732))

119902

]

= (1

1198691(120579))

119902

120579 isin [1205793 1205794]

(A21)

Conflict of Interests

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

Acknowledgments

The authors would like to thank the Academic Edi-tor Antonino Laudani and an anonymous referee fortheir insightful comments and suggestions on this paperwhich have led to significant improvements This workwas supported by the Postdoctoral Science Foundation ofChina (2014M560317) the National Science Fund of China(11271135 11471119 11371142 and 11101156) the FundamentalResearch Funds for the Central Universities the 111 Project(B14019) and the Programof Shanghai Subject Chief Scientist(14XD1401600)

References

[1] J J Goeman A Solari and T Stijnen ldquoThree-sided hypothesistesting simultaneous testing of superiority equivalence andinferiorityrdquo Statistics in Medicine vol 29 no 20 pp 2117ndash21252010

[2] K S Fu Sequential Methods in Pattern Recognition and Learn-ing Academic Press New York NY USA 1968

[3] T McMillen and P Holmes ldquoThe dynamics of choice amongmultiple alternativesrdquo Journal of Mathematical Psychology vol50 no 1 pp 30ndash57 2006

[4] J J Bussgang ldquoSequential methods in radar detectionrdquo Proceed-ings of the IEEE vol 58 no 5 pp 731ndash743 1970

[5] S L Anderson ldquoSimple method of comparing the breakingstrength of two yarnsrdquo Journal of the Textle Institute vol 45 pp472ndash479 1954

[6] Y Li X L Pu and F Tsung ldquoAdaptive charting schemesbased on double sequential probability ratio testsrdquo Quality andReliability Engineering International vol 25 no 1 pp 21ndash392009

[7] I V Pavlov ldquoA sequential procedure for testingmany compositehypothesesrdquoTheory of Probability amp Its Applications vol 32 no1 pp 138ndash142 1988

[8] C W Baum and V V Veeravalli ldquoA sequential procedurefor multihypothesis testingrdquo IEEE Transactions on InformationTheory vol 40 no 6 pp 1994ndash2007 1994

[9] V P Dragalin A G Tartakovsky and V V Veeravalli ldquoMul-tihypothesis sequential probability ratio tests I asymptoticoptimalityrdquo IEEE Transactions on Information Theory vol 45no 7 pp 2448ndash2461 1999

[10] V P Dragalin A G Tartakovsky and V V Veeravalli ldquoMul-tihypothesis sequential probability ratio tests II Accurateasymptotic expansions for the expected sample sizerdquo IEEETransactions on Information Theory vol 46 no 4 pp 1366ndash1383 2000

[11] M Sobel and A Wald ldquoA sequential decision procedure forchoosing one of three hypotheses concerning the unknownmean of a normal distributionrdquo Annals of Mathematical Statis-tics vol 20 pp 502ndash522 1949

[12] P Armitage ldquoSequential analysis withmore than two alternativehypotheses and its relation to discriminant function analysisrdquoJournal of the Royal Statistical Society Series B Methodologicalvol 12 pp 137ndash144 1950

[13] G Simons ldquoLower bounds for average sample number ofsequential multihypothesis testsrdquo Annals of MathematicalStatistics vol 38 pp 1343ndash1364 1967

8 Mathematical Problems in Engineering

[14] G Lorden ldquoLikelihood ratio tests for sequential k-decisionproblemsrdquo Annals of Mathematical Statistics vol 43 pp 1412ndash1427 1972

[15] J Whitehead and H Brunier ldquoThe double triangular test asequential test for the two-sided alternative with early stoppingunder the null hypothesisrdquo Sequential Analysis Design Methodsamp Applications vol 9 no 2 pp 117ndash136 1990

[16] Y Li and X Pu ldquoHypothesis designs for three-hypothesis testproblemsrdquo Mathematical Problems in Engineering vol 2010Article ID 393095 15 pages 2010

[17] Y Li and X L Pu ldquoA method for designing three-hypothesistest problems and sequential schemesrdquo Communications inStatisticsmdashSimulation and Computation vol 39 no 9 pp 1690ndash1708 2010

[18] V P Dragalin and A A Novikov ldquoAdaptive sequential testsfor composite hypothesesrdquo Surveys of Applied and IndustrialMathematics vol 6 pp 387ndash398 1999

[19] T L Lai ldquoSequential multiple hypothesis testing and efficientfault detection-isolation in stochastic systemsrdquo IEEE Transac-tions on Information Theory vol 46 no 2 pp 595ndash608 2000

[20] L Wang D D Xiang X L Pu and Y Li ldquoA double sequen-tial weighted probability ratio test for one-sided compositehypothesesrdquo Communications in StatisticsTheory andMethodsvol 42 no 20 pp 3678ndash3695 2013

[21] G Lorden ldquo2-SPRTrsquos and the modified Kiefer-Weiss problemof minimizing an expected sample sizerdquoTheAnnals of Statisticsvol 4 no 2 pp 281ndash291 1976

[22] J D Chen and F J Hickernell ldquoA class of asymptotically optimalsequential tests for composite hypothesesrdquo Science in ChinaSeries A vol 37 no 11 pp 1314ndash1324 1994

[23] W Hoeffding ldquoLower bounds for the expected sample sizeand the average risk of a sequential procedurerdquo Annals ofMathematical Statistics vol 31 pp 352ndash368 1960

[24] J D Chen ldquoAsymptotic optimality for one class of truncatedsequential testsrdquo Science in China Series A MathematicsPhysics Astronomy vol 30 pp 30ndash41 1991

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 Mathematical Problems in Engineering

the same results as the Sobel-Wald test when the true param-eter does not belong to Θ the proposed test also performsbetter than the Whitehead-Brunier test and has comparableperformance with the Sobel-Wald test (3) the proposed testis easy to implement and can be extended to multihypothesistesting problems Future work will be concerned with themethod of determining the boundaries in an analytical wayinstead of the Monte Carlo method

Appendix

We provide sketch proofs of Theorems 1 2 4 and 5

Proof ofTheorem 1 LetF119899= 120590(119909

1 119909

119899) 119899 = 1 2 Note

that (1198771119899 119865119899 119899 ge 1) is a supermartingale under 119875

120579 forall120579 isin Θ

1

Therefore for all 120579 isin Θ1

119875120579(119889 = 2) le 119875

120579(1205911lt infin)

le int1205911ltinfin

119860minus1

11198771

1205911119889119875120579

le 119864120579[119860minus1

11198771

1]

(A1)

On the other hand following Lemma 1 of Chen and Hicker-nell [22] for any positive integer119898 and 120579 le 120579

1lt 120582 we have

119864120579[119903119898(120582 1205791)] le 1 (A2)

Thus

119864120579[119860minus1

11198771

1] = 119860

minus1

1int

1205792

120579lowast

1

119864120579[1199031(119905 1205791)] 119892 (119905) 119889119905

le 119860minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905

(A3)

Combining (A1) and (A3) we have

119875120579(119889 = 2) le 119860

minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905 forall120579 isin Θ1 (A4)

In particular setting

1198601= 120572minus1

1int

1205792

120579lowast

1

119892 (119905) 119889119905 (A5)

we have sup120579isinΘ1

119875120579(119889 = 2) le 120572

1 Similarly we can prove

that sup120579isinΘ2

119875120579(119889 = 1) le 120573

1 sup120579isinΘ2

119875120579(119889 = 3) le 120572

2 and

sup120579isinΘ3

119875120579(119889 = 2) le 120573

2with 119861

1= 120573minus1

1int120579lowast

1

1205791

119892(119905)119889119905 1198602=

120572minus1

2int1205794

120579lowast

2

119892(119905)119889119905 and 1198612= 120573minus1

2int120579lowast

2

1205793

119892(119905)119889119905 respectively

Proof of Theorem 2 If 119892(120579) is a sectionally continuous func-tion according to Theorem 32 of Wang et al [20] we knowthat (1) for all119860

1gt 0 and 119904

1gt (120595(120579

lowast

1)minus120595(120579

1))(120579lowast

1minus1205791) there

exists 1198791198601(1199041) lt infin such that 1198771

119899ge 1198601when 119899 ge 119879

1198601(1199041) and

119878119899ge 1198991199041 (2) for all 119861

1gt 0 and 119904

1lt (120595(120579

2)minus120595(120579

lowast

1))(1205792minus120579lowast

1)

there exists 1198791198611(1199041) lt infin such that 1

119899ge 1198611when 119899 ge 119879

1198611(1199041)

and 119878119899le 1198991199041 where 119878

119899= sum119899

119897=1119909119897

Noting that120595(120579) is convex we have (120595(120579lowast1)minus120595(120579

1))(120579lowast

1minus

1205791) lt (120595(120579

2) minus 120595(120579

lowast

1))(1205792minus 120579lowast

1) It is easy to choose 119904

1such

that

120595 (120579lowast

1) minus 120595 (120579

1)

120579lowast

1minus 1205791

lt 1199041lt120595 (1205792) minus 120595 (120579

lowast

1)

1205792minus 120579lowast

1

(A6)

Let 11987911986011198611

= max(1198791198601(1199041) 1198791198611(1199041)) Then we have 120591lowast

1le 11987911986011198611

Similarly for all 119860

2gt 0 and 119861

2gt 0 we can prove that there

exist 1198791198602(1199042) lt infin and 119879

1198612(1199042) lt infin such that 120591lowast

2le 11987911986021198612

Thus we have

120591 le max (11987911986011198611

11987911986021198612

) (A7)

Proof of Theorem 4 Using Hoeffding inequality (see Hoeffd-ing [23]) we know

lim inf1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)ge 1198691(120579) 120579 isin [120579

1 1205792] (A8)

so it suffices to show

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)le 1198691(120579) 120579 isin [120579

1 1205792] (A9)

According to Theorem 33 of Wang et al [20] for all 120579 isin

[120579lowast

1 1205792]

lim1205721rarr0

1205911

log (1198601)=

1

119870 (120579 1205791)

(as 119875120579) (A10)

Since 1198601= 120572minus1

1 when 120572

1+ 1205731rarr 0 and log(120572

1) asymp log(120573

1)

we have

minus log (1205721+ 1205731) 997888rarr minus log (120572

1) = log (119860

1) (A11)

Therefore for all 120579 isin [120579lowast1 1205792]

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591lowast

1

minus log (1205721+ 1205731)le lim1205721rarr0

1198641205791205911

log (1198601)=

1

119870 (120579 1205791)

(A12)

Similarly for all 120579 isin [1205791 120579lowast

1] we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591lowast

1

minus log (1205721+ 1205731)le lim1205731rarr0

1198641205791205911

log (1198611)=

1

119870 (120579 1205792)

(A13)

Combining two inequalities (A12) and (A13) we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)le 1198691(120579) 120579 isin [120579

1 1205792] (A14)

According to (A8) and (A14)

lim1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579120591

minus log (1205721+ 1205731)= 1198691(120579) 120579 isin [120579

1 1205792] (A15)

Mathematical Problems in Engineering 7

Similarly we can prove that

lim1205722+1205732rarr0

log(1205722)asymplog(1205732)

119864120579120591

minus log (1205722+ 1205732)= 1198692(120579) 120579 isin [120579

3 1205794] (A16)

Proof of Theorem 5 Using Lemma 36 of Chen [24] for all119902 ge 1 we know

lim1205721rarr0

119864120579[(

1205911

minus log (1205721))

119902

] =1

[119870 (120579 1205791)]119902 (120579

lowast

1le 120579 le 120579

2)

lim1205731rarr0

119864120579[(

1205911

minus log (1205731))

119902

] =1

[119870 (120579 1205792)]119902 (120579

1le 120579 le 120579

lowast

1)

(A17)

Similar to Theorem 4 for all 120579 isin [120579lowast1 1205792] we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591lowast

1

minus log (1205721+ 1205731))]

119902

le lim1205721rarr0

119864120579[(

1205911

minus log (1205721))]

119902

=1

[119870 (120579 1205791)]119902

(A18)

For all 120579 isin [1205791 120579lowast

1] there is

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591lowast

1

minus log (1205721+ 1205731))]

119902

le lim1205731rarr0

119864120579[(

1205911

minus log (1205731))]

119902

=1

[119870 (120579 1205792)]119902

(A19)

According to (A18) (A19) and Hoeffding inequality wehave

lim1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591

minus log (1205721+ 1205731))

119902

]

= (1

1198691(120579))

119902

120579 isin [1205791 1205792]

(A20)

Similarly we can prove that

lim1205722+1205732rarr0

log(1205722)asymplog(1205732)

119864120579[(

120591

minus log (1205722+ 1205732))

119902

]

= (1

1198691(120579))

119902

120579 isin [1205793 1205794]

(A21)

Conflict of Interests

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

Acknowledgments

The authors would like to thank the Academic Edi-tor Antonino Laudani and an anonymous referee fortheir insightful comments and suggestions on this paperwhich have led to significant improvements This workwas supported by the Postdoctoral Science Foundation ofChina (2014M560317) the National Science Fund of China(11271135 11471119 11371142 and 11101156) the FundamentalResearch Funds for the Central Universities the 111 Project(B14019) and the Programof Shanghai Subject Chief Scientist(14XD1401600)

References

[1] J J Goeman A Solari and T Stijnen ldquoThree-sided hypothesistesting simultaneous testing of superiority equivalence andinferiorityrdquo Statistics in Medicine vol 29 no 20 pp 2117ndash21252010

[2] K S Fu Sequential Methods in Pattern Recognition and Learn-ing Academic Press New York NY USA 1968

[3] T McMillen and P Holmes ldquoThe dynamics of choice amongmultiple alternativesrdquo Journal of Mathematical Psychology vol50 no 1 pp 30ndash57 2006

[4] J J Bussgang ldquoSequential methods in radar detectionrdquo Proceed-ings of the IEEE vol 58 no 5 pp 731ndash743 1970

[5] S L Anderson ldquoSimple method of comparing the breakingstrength of two yarnsrdquo Journal of the Textle Institute vol 45 pp472ndash479 1954

[6] Y Li X L Pu and F Tsung ldquoAdaptive charting schemesbased on double sequential probability ratio testsrdquo Quality andReliability Engineering International vol 25 no 1 pp 21ndash392009

[7] I V Pavlov ldquoA sequential procedure for testingmany compositehypothesesrdquoTheory of Probability amp Its Applications vol 32 no1 pp 138ndash142 1988

[8] C W Baum and V V Veeravalli ldquoA sequential procedurefor multihypothesis testingrdquo IEEE Transactions on InformationTheory vol 40 no 6 pp 1994ndash2007 1994

[9] V P Dragalin A G Tartakovsky and V V Veeravalli ldquoMul-tihypothesis sequential probability ratio tests I asymptoticoptimalityrdquo IEEE Transactions on Information Theory vol 45no 7 pp 2448ndash2461 1999

[10] V P Dragalin A G Tartakovsky and V V Veeravalli ldquoMul-tihypothesis sequential probability ratio tests II Accurateasymptotic expansions for the expected sample sizerdquo IEEETransactions on Information Theory vol 46 no 4 pp 1366ndash1383 2000

[11] M Sobel and A Wald ldquoA sequential decision procedure forchoosing one of three hypotheses concerning the unknownmean of a normal distributionrdquo Annals of Mathematical Statis-tics vol 20 pp 502ndash522 1949

[12] P Armitage ldquoSequential analysis withmore than two alternativehypotheses and its relation to discriminant function analysisrdquoJournal of the Royal Statistical Society Series B Methodologicalvol 12 pp 137ndash144 1950

[13] G Simons ldquoLower bounds for average sample number ofsequential multihypothesis testsrdquo Annals of MathematicalStatistics vol 38 pp 1343ndash1364 1967

8 Mathematical Problems in Engineering

[14] G Lorden ldquoLikelihood ratio tests for sequential k-decisionproblemsrdquo Annals of Mathematical Statistics vol 43 pp 1412ndash1427 1972

[15] J Whitehead and H Brunier ldquoThe double triangular test asequential test for the two-sided alternative with early stoppingunder the null hypothesisrdquo Sequential Analysis Design Methodsamp Applications vol 9 no 2 pp 117ndash136 1990

[16] Y Li and X Pu ldquoHypothesis designs for three-hypothesis testproblemsrdquo Mathematical Problems in Engineering vol 2010Article ID 393095 15 pages 2010

[17] Y Li and X L Pu ldquoA method for designing three-hypothesistest problems and sequential schemesrdquo Communications inStatisticsmdashSimulation and Computation vol 39 no 9 pp 1690ndash1708 2010

[18] V P Dragalin and A A Novikov ldquoAdaptive sequential testsfor composite hypothesesrdquo Surveys of Applied and IndustrialMathematics vol 6 pp 387ndash398 1999

[19] T L Lai ldquoSequential multiple hypothesis testing and efficientfault detection-isolation in stochastic systemsrdquo IEEE Transac-tions on Information Theory vol 46 no 2 pp 595ndash608 2000

[20] L Wang D D Xiang X L Pu and Y Li ldquoA double sequen-tial weighted probability ratio test for one-sided compositehypothesesrdquo Communications in StatisticsTheory andMethodsvol 42 no 20 pp 3678ndash3695 2013

[21] G Lorden ldquo2-SPRTrsquos and the modified Kiefer-Weiss problemof minimizing an expected sample sizerdquoTheAnnals of Statisticsvol 4 no 2 pp 281ndash291 1976

[22] J D Chen and F J Hickernell ldquoA class of asymptotically optimalsequential tests for composite hypothesesrdquo Science in ChinaSeries A vol 37 no 11 pp 1314ndash1324 1994

[23] W Hoeffding ldquoLower bounds for the expected sample sizeand the average risk of a sequential procedurerdquo Annals ofMathematical Statistics vol 31 pp 352ndash368 1960

[24] J D Chen ldquoAsymptotic optimality for one class of truncatedsequential testsrdquo Science in China Series A MathematicsPhysics Astronomy vol 30 pp 30ndash41 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Similarly we can prove that

lim1205722+1205732rarr0

log(1205722)asymplog(1205732)

119864120579120591

minus log (1205722+ 1205732)= 1198692(120579) 120579 isin [120579

3 1205794] (A16)

Proof of Theorem 5 Using Lemma 36 of Chen [24] for all119902 ge 1 we know

lim1205721rarr0

119864120579[(

1205911

minus log (1205721))

119902

] =1

[119870 (120579 1205791)]119902 (120579

lowast

1le 120579 le 120579

2)

lim1205731rarr0

119864120579[(

1205911

minus log (1205731))

119902

] =1

[119870 (120579 1205792)]119902 (120579

1le 120579 le 120579

lowast

1)

(A17)

Similar to Theorem 4 for all 120579 isin [120579lowast1 1205792] we have

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591lowast

1

minus log (1205721+ 1205731))]

119902

le lim1205721rarr0

119864120579[(

1205911

minus log (1205721))]

119902

=1

[119870 (120579 1205791)]119902

(A18)

For all 120579 isin [1205791 120579lowast

1] there is

lim sup1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591lowast

1

minus log (1205721+ 1205731))]

119902

le lim1205731rarr0

119864120579[(

1205911

minus log (1205731))]

119902

=1

[119870 (120579 1205792)]119902

(A19)

According to (A18) (A19) and Hoeffding inequality wehave

lim1205721+1205731rarr0

log(1205721)asymplog(1205731)

119864120579[(

120591

minus log (1205721+ 1205731))

119902

]

= (1

1198691(120579))

119902

120579 isin [1205791 1205792]

(A20)

Similarly we can prove that

lim1205722+1205732rarr0

log(1205722)asymplog(1205732)

119864120579[(

120591

minus log (1205722+ 1205732))

119902

]

= (1

1198691(120579))

119902

120579 isin [1205793 1205794]

(A21)

Conflict of Interests

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

Acknowledgments

The authors would like to thank the Academic Edi-tor Antonino Laudani and an anonymous referee fortheir insightful comments and suggestions on this paperwhich have led to significant improvements This workwas supported by the Postdoctoral Science Foundation ofChina (2014M560317) the National Science Fund of China(11271135 11471119 11371142 and 11101156) the FundamentalResearch Funds for the Central Universities the 111 Project(B14019) and the Programof Shanghai Subject Chief Scientist(14XD1401600)

References

[1] J J Goeman A Solari and T Stijnen ldquoThree-sided hypothesistesting simultaneous testing of superiority equivalence andinferiorityrdquo Statistics in Medicine vol 29 no 20 pp 2117ndash21252010

[2] K S Fu Sequential Methods in Pattern Recognition and Learn-ing Academic Press New York NY USA 1968

[3] T McMillen and P Holmes ldquoThe dynamics of choice amongmultiple alternativesrdquo Journal of Mathematical Psychology vol50 no 1 pp 30ndash57 2006

[4] J J Bussgang ldquoSequential methods in radar detectionrdquo Proceed-ings of the IEEE vol 58 no 5 pp 731ndash743 1970

[5] S L Anderson ldquoSimple method of comparing the breakingstrength of two yarnsrdquo Journal of the Textle Institute vol 45 pp472ndash479 1954

[6] Y Li X L Pu and F Tsung ldquoAdaptive charting schemesbased on double sequential probability ratio testsrdquo Quality andReliability Engineering International vol 25 no 1 pp 21ndash392009

[7] I V Pavlov ldquoA sequential procedure for testingmany compositehypothesesrdquoTheory of Probability amp Its Applications vol 32 no1 pp 138ndash142 1988

[8] C W Baum and V V Veeravalli ldquoA sequential procedurefor multihypothesis testingrdquo IEEE Transactions on InformationTheory vol 40 no 6 pp 1994ndash2007 1994

[9] V P Dragalin A G Tartakovsky and V V Veeravalli ldquoMul-tihypothesis sequential probability ratio tests I asymptoticoptimalityrdquo IEEE Transactions on Information Theory vol 45no 7 pp 2448ndash2461 1999

[10] V P Dragalin A G Tartakovsky and V V Veeravalli ldquoMul-tihypothesis sequential probability ratio tests II Accurateasymptotic expansions for the expected sample sizerdquo IEEETransactions on Information Theory vol 46 no 4 pp 1366ndash1383 2000

[11] M Sobel and A Wald ldquoA sequential decision procedure forchoosing one of three hypotheses concerning the unknownmean of a normal distributionrdquo Annals of Mathematical Statis-tics vol 20 pp 502ndash522 1949

[12] P Armitage ldquoSequential analysis withmore than two alternativehypotheses and its relation to discriminant function analysisrdquoJournal of the Royal Statistical Society Series B Methodologicalvol 12 pp 137ndash144 1950

[13] G Simons ldquoLower bounds for average sample number ofsequential multihypothesis testsrdquo Annals of MathematicalStatistics vol 38 pp 1343ndash1364 1967

8 Mathematical Problems in Engineering

[14] G Lorden ldquoLikelihood ratio tests for sequential k-decisionproblemsrdquo Annals of Mathematical Statistics vol 43 pp 1412ndash1427 1972

[15] J Whitehead and H Brunier ldquoThe double triangular test asequential test for the two-sided alternative with early stoppingunder the null hypothesisrdquo Sequential Analysis Design Methodsamp Applications vol 9 no 2 pp 117ndash136 1990

[16] Y Li and X Pu ldquoHypothesis designs for three-hypothesis testproblemsrdquo Mathematical Problems in Engineering vol 2010Article ID 393095 15 pages 2010

[17] Y Li and X L Pu ldquoA method for designing three-hypothesistest problems and sequential schemesrdquo Communications inStatisticsmdashSimulation and Computation vol 39 no 9 pp 1690ndash1708 2010

[18] V P Dragalin and A A Novikov ldquoAdaptive sequential testsfor composite hypothesesrdquo Surveys of Applied and IndustrialMathematics vol 6 pp 387ndash398 1999

[19] T L Lai ldquoSequential multiple hypothesis testing and efficientfault detection-isolation in stochastic systemsrdquo IEEE Transac-tions on Information Theory vol 46 no 2 pp 595ndash608 2000

[20] L Wang D D Xiang X L Pu and Y Li ldquoA double sequen-tial weighted probability ratio test for one-sided compositehypothesesrdquo Communications in StatisticsTheory andMethodsvol 42 no 20 pp 3678ndash3695 2013

[21] G Lorden ldquo2-SPRTrsquos and the modified Kiefer-Weiss problemof minimizing an expected sample sizerdquoTheAnnals of Statisticsvol 4 no 2 pp 281ndash291 1976

[22] J D Chen and F J Hickernell ldquoA class of asymptotically optimalsequential tests for composite hypothesesrdquo Science in ChinaSeries A vol 37 no 11 pp 1314ndash1324 1994

[23] W Hoeffding ldquoLower bounds for the expected sample sizeand the average risk of a sequential procedurerdquo Annals ofMathematical Statistics vol 31 pp 352ndash368 1960

[24] J D Chen ldquoAsymptotic optimality for one class of truncatedsequential testsrdquo Science in China Series A MathematicsPhysics Astronomy vol 30 pp 30ndash41 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

[14] G Lorden ldquoLikelihood ratio tests for sequential k-decisionproblemsrdquo Annals of Mathematical Statistics vol 43 pp 1412ndash1427 1972

[15] J Whitehead and H Brunier ldquoThe double triangular test asequential test for the two-sided alternative with early stoppingunder the null hypothesisrdquo Sequential Analysis Design Methodsamp Applications vol 9 no 2 pp 117ndash136 1990

[16] Y Li and X Pu ldquoHypothesis designs for three-hypothesis testproblemsrdquo Mathematical Problems in Engineering vol 2010Article ID 393095 15 pages 2010

[17] Y Li and X L Pu ldquoA method for designing three-hypothesistest problems and sequential schemesrdquo Communications inStatisticsmdashSimulation and Computation vol 39 no 9 pp 1690ndash1708 2010

[18] V P Dragalin and A A Novikov ldquoAdaptive sequential testsfor composite hypothesesrdquo Surveys of Applied and IndustrialMathematics vol 6 pp 387ndash398 1999

[19] T L Lai ldquoSequential multiple hypothesis testing and efficientfault detection-isolation in stochastic systemsrdquo IEEE Transac-tions on Information Theory vol 46 no 2 pp 595ndash608 2000

[20] L Wang D D Xiang X L Pu and Y Li ldquoA double sequen-tial weighted probability ratio test for one-sided compositehypothesesrdquo Communications in StatisticsTheory andMethodsvol 42 no 20 pp 3678ndash3695 2013

[21] G Lorden ldquo2-SPRTrsquos and the modified Kiefer-Weiss problemof minimizing an expected sample sizerdquoTheAnnals of Statisticsvol 4 no 2 pp 281ndash291 1976

[22] J D Chen and F J Hickernell ldquoA class of asymptotically optimalsequential tests for composite hypothesesrdquo Science in ChinaSeries A vol 37 no 11 pp 1314ndash1324 1994

[23] W Hoeffding ldquoLower bounds for the expected sample sizeand the average risk of a sequential procedurerdquo Annals ofMathematical Statistics vol 31 pp 352ndash368 1960

[24] J D Chen ldquoAsymptotic optimality for one class of truncatedsequential testsrdquo Science in China Series A MathematicsPhysics Astronomy vol 30 pp 30ndash41 1991

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