research article asymptotic optimality of combined double sequential weighted...
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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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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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
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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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
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
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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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
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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