# Asymptotic Optimality of Combined Double Sequential Weighted Probability Ratio Test for Three Composite Hypotheses

Authors: Lei Wang; Xiaolong Pu; Yan Li
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

We propose the weighted expected sample size (WESS) to evaluate the overall performance on the indifference-zones for three composite hypotheses’ testing problem. Based on minimizing the WESS to control the expected sample sizes, a new sequential test is developed by utilizing two double sequential weighted probability ratio tests (2-SWPRTs) simultaneously. It is proven that the proposed test has a finite stopping time and is asymptotically optimal in the sense of asymptotically minimizing not only the expected sample size but also any positive moment of the stopping time on the indifference-zones under some mild conditions. Simulation studies illustrate that the proposed test has the smallest WESS and relative mean index (RMI) compared with Sobel-Wald and Whitehead-Brunier tests.


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## Body

## 1. Introduction

LetX1,X2,… be independent and identically distributed (i.i.d.) random variables whose common density function f(x,θ) (with respect to some nondegenerate measure ν) belongs to the exponential family (1)fx,θ=exp⁡xθ-ψθ,θ∈Θ=θ_,θ¯,where ψ(·) is a convex function and Θ is the natural parameter space with -∞≤θ_<θ¯≤∞. The problem of interest is the following three composite hypotheses’ testing problem: (2)H1:θ≤θ1versusH2:θ2≤θ≤θ3versusH3:θ≥θ4θ1<θ2<θ3<θ4,where θ1,θ2,θ3,θ4∈Θ. For example, in clinical trial applications, in order to compare the effects of two drugs (Goeman et al. [1]), the equivalence trial H0:Δ>Δ1 versus H1:Δ<Δ0(0<Δ0<Δ1) would be more realistically stated as H1:Δ<-Δ1 (inferiority), H2:-Δ0<Δ<Δ0 (equivalence), and H3:Δ>Δ1 (superiority), where Δ is the difference of effect between two drugs. The sequential testing of three or more hypotheses has been applied to a variety of engineering problems such as pattern recognition (Fu [2]; McMillen and Holmes [3]), multiple-resolution radar detection (Bussgang [4]), products comparisons (Anderson [5]), and others (Li et al. [6]). The intervals of [θ1,θ2] and [θ3,θ4] are usually called indifference-zones and denoted by Θ¯=θ1,θ2∪θ3,θ4.Published work on this problem has taken two main approaches. Pavlov [7], Baum and Veeravalli [8], and Dragalin et al. [9, 10] studied the class of tests motivated by the Bayesian framework. The second approach has focused on extending the sequential probability ratio test (SPRT) and double sequential probability ratio test (2-SPRT) to incorporate more than two hypotheses, such as Sobel and Wald [11], Armitage [12], Simons [13], Lorden [14], Whitehead and Brunier [15], and Li and Pu [16, 17]. Dragalin and Novikov [18] studied the problem of testing several composite hypotheses with an indifference-zone for an unknown parameter. Lai [19] considered the multihypothesis testing problem where some or all of these hypotheses are composite.Among others, the tests proposed by Sobel and Wald [11] and Whitehead and Brunier [15] are usually used in practice for problem (2). Specifically, Sobel and Wald [11] proposed carrying out simultaneous SPRTs of H1 versus H2 and H2 versus H3. However, when the true parameter is in the indifference-zones, the expected sample size of the Sobel-Wald test can be considerably larger than that of a fixed-sample-size test plan. Moreover, it is untruncated such that the number of observations required can not be predetermined, an undesirable property in many practical situations such as medical trial. To reduce the maximum expected sample size, Whitehead and Brunier [15] applied two 2-SPRTs instead of two SPRTs for the component tests, at the cost of larger expected sample sizes when the true parameter does not belong to the indifference-zones.For one-sided composite hypotheses, in order to control the expected sample sizes, Wang et al. [20] proposed the double sequential weighted probability ratio test (2-SWPRT) based on mixture likelihood ratio statistics and showed that the 2-SWPRT is an asymptotically overall optimal test in the sense of asymptotically minimizing the expected sample sizes on the indifference-zone. Motivated by the attractive properties of the 2-SWPRT, we extend the existing work on problem (2) from pointwise optimality to overall performance optimality when there are different concerns of interest on different θs. In particular, we propose an optimality criterion to evaluate the overall performance of sequential test plans on the indifference-zones for three composite hypotheses and correspondingly develop a new sequential test for problem (2) by utilizing two 2-SWPRTs as the component tests to reduce the expected sample sizes. We show the proposed test has a finite stopping time and is asymptotically optimal in the sense of asymptotically minimizing not only the expected sample size but also any positive moment of the stopping time on the indifference-zones. Simulation studies show that the proposed test not only has the smallest WESS compared with Sobel-Wald and Whitehead-Brunier tests, but also is superior to the Whitehead-Brunier test and comparable with the Sobel-Wald test when the true parameter does not belong to the indifference-zones. Moreover, the RMI also shows the proposed test is an efficient method to improve the overall performance.The rest of this paper is organized as follows. In Section2, we review the Sobel-Wald and Whitehead-Brunier tests. The combined double sequential weighted probability ratio test (denoted by combined 2-SWPRT) is proposed and its properties are given in Section 3. Simulation results are provided in Section 4 and some conclusions are in Section 5. All technical details are given in Appendix.


## 2. Methodology Review

For one-sided composite hypothesesH1 versus H2, the SPRT is optimal in the sense that it minimizes the expected sample sizes at θ1 and θ2, and the 2-SPRT has (approximately) minimal maximum expected sample size over (θ1,θ2) among all sequential and nonsequential tests with the same error probabilities. Given the well-known optimality properties of the SPRT and 2-SPRT, it is natural to use the SPRTs and 2-SPRTs as the component tests to construct the sequential tests for problem (2), respectively. In this section, we briefly review the Sobel-Wald and Whitehead-Brunier tests.For testing problem (2), the generalization of errors of types I and II is expressible in terms of a 3 × 3 error matrix E=(αjk), where αjk=PacceptingHj∣Hkistrue for j,k=1,2,3. However, under some mild conditions, Sobel and Wald [11, pages 504-505] and Armitage [12, pages 142-143] showed that α31 and α13 are zero, which can be verified by the simulation results in Section 4. It becomes apparent that in the general case we have at most four “degrees of freedom” in choosing an error matrix. Without loss of generality, we consider Δ=(τ,d) as a sequential test for problem (2), where τ is the stopping rule and d is the decision rule (d=k means accepting Hk, k=1,2,3). Set Θ1=θ_,θ1, Θ2=θ2,θ3 and Θ3=θ4,θ¯. Given positive vectors α=(α1,α2) and β=(β1,β2) (αi+βi<1, i=1,2),(3)Υα,β=Δ=τ,d:sup⁡θ∈ΘiPd=i+1≤αi,sup⁡θ∈Θi+1Pd=i≤βi,i=1,2is the set of all sequential tests with error probabilities controlled by α and β.(1) Sobel-Wald Test. Since the hypotheses H1, H2, and H3 are ordered, the sequential testing of problem (2) can be constructed by combining the following two one-sided composite hypotheses S1 and S2:(4)S1:θ≤θ1versusθ2≤θ≤θ3S2:θ2≤θ≤θ3versusθ≥θ4.Sobel and Wald [11] proposed operating S1 and S2 by the SPRTs simultaneously. For all θ,ρ∈Θ, define rn(θ,ρ)=∏l=1nf(xl,θ)/f(xl,ρ). The stopping and decision rules of Si(i=1,2) determined by the SRPT are(5)τis=inf⁡n≥1:rnθ2i,θ2i-1∉Ais,Bis,i=1,2,dis=Iτis<∞,rτisθ2i,θ2i-1≥Bis+i,where I(·) is the indicator function and Ais and Bis(i=1,2) are the boundary parameters (0<Ais<1<Bis<∞), which are usually set as (6)Ais=βi1-αi,Bis=1-βiαi,i=1,2,to meet requirements on the error probabilities. When B1s/B2s≤1 and A1s/A2s≤1, Sobel and Wald [11] showed the event {d1s=1,d2s=3} is impossible. The stopping and decision rules of the Sobel-Wald test are defined as (7)τs=max⁡τ1s,τ2s,ds=1,d1s=1,d2s=2;2,d1s=2,d2s=2;3,d1s=2,d2s=3.The Sobel-Wald test is optimal in the sense that it minimizes the expected sample sizes at θ2i-1 and θ2i(i=1,2) among all sequential and nonsequential tests whose error probabilities satisfy Υ(α,β). However, its expected sample sizes at other parameters over Θ¯ may be unsatisfactory.(2) Whitehead-Brunier Test. In order to minimize the maximum expected sample size under constraints (3), Whitehead and Brunier [15] applied the 2-SPRT to operate S1 and S2, instead of the SPRT. As in Lorden [21], let K(θ,ρ)=Eθlog⁡[f(x,θ)/f(x,ρ)] be the Kullback-Leibler (KL) information number. Define θ^i∈(θ2i-1,θ2i) and ni∗(i=1,2) by(8)log⁡αiKθ^i,θ2i-1=log⁡βiKθ^i,θ2i=ni∗,i=1,2.Set ci∗ such that Φ(ci∗)=-a2i∗/(a2i-1∗-a2i∗), i=1,2, where Φ(·) is the cumulative distribution function of the standard normal distribution, a2i-1∗=(θ^i-θ2i-1)/K(θ^i,θ2i-1), and a2i∗=(θ^i-θ2i)/K(θ^i,θ2i), i=1,2. Let(9)θi∗=θ^i+ci∗ni∗ψ′′θ^i-1/2,i=1,2.The stopping and decision rules of Si(i=1,2) determined by the 2-SPRT are (10)τiw=inf⁡n≥1:rnθi∗,θ2i-1≥Aiworrnθi∗,θ2i≥Biw,i=1,2,diw=Iτiw<∞,rτiwθi∗,θ2i-1≥Aiw+i,where Aiw and Biw(i=1,2) are the boundary parameters (0<Aiw,Biw<∞). The conservative values of Aiw and Biw are 1/αi and 1/βi, in the sense that the real error probabilities may be much smaller than αi and βi(i=1,2), respectively. The stopping and decision rules of the Whitehead-Brunier test are defined as (11)τw=max⁡τ1w,τ2w,dw=1,d1w=1,d2w=2;2,d1w=2,d2w=2;3,d1w=2,d2w=3.


## 3. Optimality Criterion and Combined 2-SWPRT

For testing problem (2), if θ<θ1 we prefer to accept H1 and this preference is the stronger the smaller θ. Similarly, if θ>θ4 we prefer to accept H3, and we prefer to accept H2 if θ2<θ<θ3. However, we have no strong preference between H1 and H2 if θ∈[θ1,θ2], and we also have no strong preference between H2 and H3 if θ∈[θ3,θ4]. In these cases, we need more observations for decision. Thus, when the error probabilities satisfy Υ(α,β), we focus on reduction of the expected sample sizes over the indifference-zones Θ¯ in applications. Let g(θ) be a nonnegative weight function which is sectionally continuous on [θ1,θ2] and [θ3,θ4], respectively, and satisfies ∫Θ¯‍g(θ)dθ=1. We define the weighted expected sample size as (12)WESSg=∫Θ¯‍Eθτ·gθdθto evaluate the overall performance of sequential test plans on Θ¯. The choice of g should be chosen according to practical needs (Sobel and Wald [11]). For example, let g(θ) be uniform weights when there are no differences on Θ¯; let g(θ) be assigned more weights when we focus more on reducing the expected sample size on these parameter points. As an overall evaluation, the WESS(g) integrates the performances on Θ¯ by weighting the expected sample sizes.Motivated by Wang et al. [20], we propose operating S1 and S2 by the 2-SWPRT. Specifically, the stopping and decision rules of Si(i=1,2) by the 2-SWPRT are (13)τi∗=inf⁡n≥1:Rni≥AiorR~ni≥Bi,i=1,2,di=Iτi∗<∞,Rτi∗i≥Ai+i,where(14)Rni=∫θi∗θ2i‍rnθ,θ2i-1gθdθ,R~ni=∫θ2i-1θi∗‍rnθ,θ2igθdθ,i=1,2,where Ai and Bi(i=1,2) are the boundary parameters (0<Ai,Bi<∞). Hence, the stopping and decision rules of the combined 2-SWPRT are defined as (15)τ=max⁡τ1∗,τ2∗,d=1,d1=1,d2=2;2,d1=2,d2=2;3,d1=2,d2=3.Some features of the combined 2-SWPRT are provided in the following theorems, whose proofs are provided in appendices.First, we show the error probabilities of the combined 2-SWPRT can be easily controlled and the stopping time is finite.Theorem 1.
There exist boundariesAi and Bi(i=1,2) such that (τ,d) in (15) belongs to Υ(α,β).Theorem 2.
For any given nonnegative sectionally continuous weight functiong(θ), the stopping time of the combined 2-SWPRT is finite.Second, we prove that the combined 2-SWPRT is asymptotically optimal onΘ¯.Definition 3.
(
τ
,
d
)
∈
Υ
(
α
,
β
) is said to be asymptotically optimal on Θ¯, if (16)limαi+βi→0log⁡⁡(αi)≈log⁡⁡(βi)Eθτ-log⁡αi+βi=Jiθ,θ∈θ2i-1,θ2i,where Ji(θ)=min⁡1/K(θ,θ2i-1),1/K(θ,θ2i), i=1,2.Theorem 4.
WhenAi=αi-1 and Bi=βi-1, the (τ,d) defined by (15) is asymptotically optimal on Θ¯.Third, we show that any positive moment of the stopping time is asymptotically optimal on the indifference-zones.Theorem 5.
Under the conditions of Theorem4, for all q≥1 and θ∈[θ2i-1,θ2i], i=1,2,(17)limαi+βi→0log⁡(αi)≈log⁡(βi)Eθτ-log⁡⁡αi+βiq=1Jiθq,i=1,2.


## 4. Simulation Studies

In this section, we conduct simulation studies to examine the performances of the combined 2-SWPRT, the Sobel-Wald test, and Whitehead-Brunier test based on the normal and Bernoulli distributions. In particular, we considered two weight functions forg(θ) as follows; (1) uniform weights: g(θ)=0.5∑i=12‍I(θ∈[θ2i-1,θ2i])/(θ2i-θ2i-1); (2) KL weights: g(θ)=0.5∑i=12‍I(θ∈[θ2i-1,θ2i])Mi(θ)/[∫θ2i-1θ2i‍Mi(θ)dθ], where Mi(θ)=max⁡(K(θ,θ2i-1),K(θ,θ2i)). As in Wang et al. [20], the corresponding formulations of the statistics Rni and R~ni(i=1,2) can be obtained. The boundaries of the tests are determined through 106 Monte Carlo trials, which make the relative differences between the real error probabilities (αi′ and βi′) and the required ones (αi and βi) within 1%; that is, |αi′-αi|/αi<1% and |βi′-βi|/βi<1%.Given the boundaries, we obtained the simulatedWESS(g)=∑θ∈S‍Eθτ·g(θ) to approximate integral (12) as follows. Let [θ1,θ2] and [θ3,θ3] be discrete as the finite sets of parameters S1=[θ1,θ1+Δ,θ1+2Δ,…,θ1∗,…,θ2-Δ,θ2] and S2=[θ3,θ3+Δ,θ3+2Δ,…,θ2∗,…,θ4-Δ,θ4] with increase Δ, respectively. Denote S=S1∪S2⊂Θ¯ and the weight function g(θ) is calculated based on θ∈S; that is, g(θ)=0.5∑i=12‍I(θ∈Si)Mi(θ)/[∑θ∈Si‍Mi(θ)dθ] for KL weights. We also compute the RMI to assess the relative efficiency between different test plans. According to Wang et al. [20], we define (18)RMIg=∑θ∈S‍Eθτ-SEθτSEθτ·gθ,where SEθτ is the smallest Eθτ among the compared tests, that is, the Sobel-Wald test, the Whitehead-Brunier test, and the combined 2-SWPRT. A test plan with a smaller RMI(g) value is considered better in its overall performance.


### 4.1. Test for the Normal Mean with Known Variance

SupposeX1,X2,… are i.i.d. from N(θ,1), -θ1=θ4=1.5, -θ2=θ3=0.5, and α1=α2=β1=β2=0.01. According to Lorden [21], we have θ1∗=-1 and θ2∗=1. The stopping boundaries are obtained as follows:(1)
for the Sobel-Wald test,Ais=0.018 and Bis=55.73;(2)
for the Whitehead-Brunier test,Aiw=Biw=37.36;(3)
for the combined 2-SWPRT,Ai=Bi=26.92 for the uniform weights and Ai=Bi=13.78 for the KL weights, respectively.As expected, we found that α13 and α31 of these three tests are equal to 0. Set Δ=0.05. Through another simulation study with 105 replications, the WESS(g) and RMI(g) are presented in Table 1. Similarly, the expected sample sizes for θ∈[-2,2] are illustrated in Figure 1.Table 1
WESS(g) and RMI(g) for testing normal mean.
g
Uniform weights
KL weights
Sobel-Wald
Whitehead-Brunier
Combined 2-SWPRT
Sobel-Wald
Whitehead-Brunier
Combined 2-SWPRT
WESS
15.231
14.342
13.511
13.818
13.734
12.694
RMI
0.268
0.1652
0.0183
0.1629
0.2117
0.0178Figure 1
Expected sample sizes for testing normal mean,α1=α2=β1=β2=0.01, -θ1=θ4=1.5, and -θ2=θ3=0.5.It is clear that the combined 2-SWPRTs have the smallestWESS(g) in all cases. In fact, compared with the Sobel-Wald and Whitehead-Brunier tests, the WESS(g) of the combined 2-SWPRT has been reduced by 11.36% and 5.86% for the uniform weights, and 8.13% and 7.57% for the KL weights. Meanwhile, in terms of the RMI(g), the combined 2-SWPRT also performs best overall.From Figure1, it also can be seen that the expected sample size of the combined 2-SWPRT is slightly larger than the Whitehead-Brunier test when the true parameter is close to θi∗ (i=1,2) and almost the same as the Sobel-Wald test when the true parameter is close to θ2i-1 or θ2i (i=1,2). When the true parameter belongs to Θk(k=1,2,3), the combined 2-SWPRT performs better than the Whitehead-Brunier test and is comparable with the Sobel-Wald test.


### 4.2. Test for the True Proportion of a Bernoulli Distribution

SupposeX1,X2,… are i.i.d. random variables from the Bernoulli distribution and P(X1=1)=p=1-P(X1=0)(0<p<1). The three composite hypotheses’ testing problem is (19)H0:p≤p1versusH1:p2≤p≤p3versusH3:p≥p4,where 0<p1<p2<p3<p4<1. Let α1=α2=β1=β2=0.01, p1=0.1, p2=0.3, p3=0.4, and p4=0.7. According to (9), we have(20)pi∗=log⁡1-p2i-1/1-p2ilog⁡p2i/p2i-1+log⁡1-p2i-1/1-p2i,such that p1∗=0.186 and p2∗=0.553 in the Whitehead-Brunier test and combined 2-SWPRT. The stopping boundaries are obtained as follows: (1)
for the Sobel-Wald test,A1s=0.012, B1s=66.52, A2s=0.014, and B2s=77.23;(2)
for the Whitehead-Brunier test,A1w=39.78, B1w=46.51, A2w=44.33, and B2w=43.27;(3)
for the combined 2-SWPRT,A1=13.06, B1=20.32, A2=16.96, and B2=16.27 for the uniform weights and A1=19.62, B1=14.35, A2=17.18, and B2=13.27 for the KL weights.In this case, the values of α13=α31=0. Set Δ=0.0625. Through another simulation study with 105 replications, the WESS(g) and RMI(g) are presented in Table 2. Similarly, the expected sample sizes for p∈(0,1) are illustrated in Figure 2.Table 2
WESS(g) and RMI(g) for testing proportion in Bernoulli distribution.
g
Uniform weights
KL weights
Sobel-Wald
Whitehead-Brunier
Combined 2-SWPRT
Sobel-Wald
Whitehead-Brunier
Combined 2-SWPRT
WESS
48.978
44.163
43.253
43.387
41.262
40.035
RMI
0.2508
0.0726
0.0163
0.1615
0.0985
0.0215Figure 2
Expected sample sizes for testing proportion in Bernoulli distribution,α1=α2=β1=β2=0.01, p2=0.1, p2=0.3, p3=0.4, and p4=0.7.It can be seen from Table2 that the combined 2-SWPRT still has the smallest WESS(g) and RMI(g) for the Bernoulli distribution. Meanwhile, from Figure 2, we have similar conclusions as those in the normal distribution cases in Section 4.1.


## 4.1. Test for the Normal Mean with Known Variance

SupposeX1,X2,… are i.i.d. from N(θ,1), -θ1=θ4=1.5, -θ2=θ3=0.5, and α1=α2=β1=β2=0.01. According to Lorden [21], we have θ1∗=-1 and θ2∗=1. The stopping boundaries are obtained as follows:(1)
for the Sobel-Wald test,Ais=0.018 and Bis=55.73;(2)
for the Whitehead-Brunier test,Aiw=Biw=37.36;(3)
for the combined 2-SWPRT,Ai=Bi=26.92 for the uniform weights and Ai=Bi=13.78 for the KL weights, respectively.As expected, we found that α13 and α31 of these three tests are equal to 0. Set Δ=0.05. Through another simulation study with 105 replications, the WESS(g) and RMI(g) are presented in Table 1. Similarly, the expected sample sizes for θ∈[-2,2] are illustrated in Figure 1.Table 1
WESS(g) and RMI(g) for testing normal mean.
g
Uniform weights
KL weights
Sobel-Wald
Whitehead-Brunier
Combined 2-SWPRT
Sobel-Wald
Whitehead-Brunier
Combined 2-SWPRT
WESS
15.231
14.342
13.511
13.818
13.734
12.694
RMI
0.268
0.1652
0.0183
0.1629
0.2117
0.0178Figure 1
Expected sample sizes for testing normal mean,α1=α2=β1=β2=0.01, -θ1=θ4=1.5, and -θ2=θ3=0.5.It is clear that the combined 2-SWPRTs have the smallestWESS(g) in all cases. In fact, compared with the Sobel-Wald and Whitehead-Brunier tests, the WESS(g) of the combined 2-SWPRT has been reduced by 11.36% and 5.86% for the uniform weights, and 8.13% and 7.57% for the KL weights. Meanwhile, in terms of the RMI(g), the combined 2-SWPRT also performs best overall.From Figure1, it also can be seen that the expected sample size of the combined 2-SWPRT is slightly larger than the Whitehead-Brunier test when the true parameter is close to θi∗ (i=1,2) and almost the same as the Sobel-Wald test when the true parameter is close to θ2i-1 or θ2i (i=1,2). When the true parameter belongs to Θk(k=1,2,3), the combined 2-SWPRT performs better than the Whitehead-Brunier test and is comparable with the Sobel-Wald test.


## 4.2. Test for the True Proportion of a Bernoulli Distribution

SupposeX1,X2,… are i.i.d. random variables from the Bernoulli distribution and P(X1=1)=p=1-P(X1=0)(0<p<1). The three composite hypotheses’ testing problem is (19)H0:p≤p1versusH1:p2≤p≤p3versusH3:p≥p4,where 0<p1<p2<p3<p4<1. Let α1=α2=β1=β2=0.01, p1=0.1, p2=0.3, p3=0.4, and p4=0.7. According to (9), we have(20)pi∗=log⁡1-p2i-1/1-p2ilog⁡p2i/p2i-1+log⁡1-p2i-1/1-p2i,such that p1∗=0.186 and p2∗=0.553 in the Whitehead-Brunier test and combined 2-SWPRT. The stopping boundaries are obtained as follows: (1)
for the Sobel-Wald test,A1s=0.012, B1s=66.52, A2s=0.014, and B2s=77.23;(2)
for the Whitehead-Brunier test,A1w=39.78, B1w=46.51, A2w=44.33, and B2w=43.27;(3)
for the combined 2-SWPRT,A1=13.06, B1=20.32, A2=16.96, and B2=16.27 for the uniform weights and A1=19.62, B1=14.35, A2=17.18, and B2=13.27 for the KL weights.In this case, the values of α13=α31=0. Set Δ=0.0625. Through another simulation study with 105 replications, the WESS(g) and RMI(g) are presented in Table 2. Similarly, the expected sample sizes for p∈(0,1) are illustrated in Figure 2.Table 2
WESS(g) and RMI(g) for testing proportion in Bernoulli distribution.
g
Uniform weights
KL weights
Sobel-Wald
Whitehead-Brunier
Combined 2-SWPRT
Sobel-Wald
Whitehead-Brunier
Combined 2-SWPRT
WESS
48.978
44.163
43.253
43.387
41.262
40.035
RMI
0.2508
0.0726
0.0163
0.1615
0.0985
0.0215Figure 2
Expected sample sizes for testing proportion in Bernoulli distribution,α1=α2=β1=β2=0.01, p2=0.1, p2=0.3, p3=0.4, and p4=0.7.It can be seen from Table2 that the combined 2-SWPRT still has the smallest WESS(g) and RMI(g) for the Bernoulli distribution. Meanwhile, from Figure 2, we have similar conclusions as those in the normal distribution cases in Section 4.1.


## 5. Summary

In this paper, we propose theWESS(g) to evaluate the overall performance on the indifference-zones for three composite hypotheses’ testing problem. In order to minimize WESS(g) to control the expected sample sizes, we developed a new sequential test by utilizing two 2-SWPRTs simultaneously. We have shown the proposed test is an asymptotically optimal test in the sense of asymptotically minimizing the expected sample sizes on the indifferent-zones.According to the simulation results, compared with the Sobel-Wald and Whitehead-Brunier tests, we conclude that the proposed test has the following merits:(1) it has the smallest WESS(g) and RMI(g); (2) when the true parameter is close to θi∗(i=1,2), the proposed test has comparable performance with Whitehead-Brunier test; when the true parameter is close to θ2i-1 or θ2i(i=1,2), it has almost the same results as the Sobel-Wald test; when the true parameter does not belong to Θ¯, the proposed test also performs better than the Whitehead-Brunier test and has comparable performance with the Sobel-Wald test; (3) the proposed test is easy to implement and can be extended to multihypothesis testing problems. Future work will be concerned with the method of determining the boundaries in an analytical way instead of the Monte Carlo method.



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