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

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 smallestWESS and relativemean index (RMI) comparedwith Sobel-Wald and Whitehead-Brunier tests.


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Mathematical Problems in Engineering 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  1 versus  2 and  2 versus  3 .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 s.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 Section 2, 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.

Methodology Review
For one-sided composite hypotheses  1 versus  2 , the SPRT is optimal in the sense that it minimizes the expected sample sizes at  1 and  2 , and the 2-SPRT has (approximately) minimal maximum expected sample size over ( 1 ,  2 ) 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  = (  ), where   = [accepting   |   is true] for ,  = 1, 2, 3.However, under some mild conditions, Sobel and Wald [11, pages 504-505] and Armitage [12, pages 142-143] showed that  31 and  13 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 = (, ) as a sequential test for problem (2), where  is the stopping rule and  is the decision rule ( =  means accepting is the set of all sequential tests with error probabilities controlled by  and . (1) Sobel-Wald Test.Since the hypotheses  1 ,  2 , and  3 are ordered, the sequential testing of problem (2) can be constructed by combining the following two one-sided composite hypotheses  1 and  2 : Sobel and Wald [11] proposed operating  1 and  2 by the SPRTs simultaneously.For all ,  ∈ Θ, define   (, ) = ∏  =1 (  , )/(  , ).The stopping and decision rules of   ( = 1, 2) determined by the SRPT are where (⋅) is the indicator function and    and    ( = 1, 2) are the boundary parameters (0 <    < 1 <    < ∞), which are usually set as to meet requirements on the error probabilities.When   1 /  2 ≤ 1 and   1 /  2 ≤ 1, Sobel and Wald [11] showed the event {  1 = 1,   2 = 3} is impossible.The stopping and decision rules of the Sobel-Wald test are defined as The Sobel-Wald test is optimal in the sense that it minimizes the expected sample sizes at  2−1 and  2 ( = 1, 2) among all sequential and nonsequential tests whose error probabilities satisfy Υ(, ).However, its expected sample sizes at other parameters over Θ may be unsatisfactory. ( Set The stopping and decision rules of   ( = 1, 2) determined by the 2-SPRT are where    and    ( = 1, 2) are the boundary parameters (0 <    ,    < ∞).The conservative values of    and    are 1/  and 1/  , in the sense that the real error probabilities may be much smaller than   and   ( = 1, 2), respectively.The stopping and decision rules of the Whitehead-Brunier test are defined as

Optimality Criterion and Combined 2-SWPRT
to evaluate the overall performance of sequential test plans on Θ.The choice of  should be chosen according to practical needs (Sobel and Wald [11]).For example, let () be uniform weights when there are no differences on Θ; let () be assigned more weights when we focus more on reducing the expected sample size on these parameter points.As an overall evaluation, the WESS() integrates the performances on Θ by weighting the expected sample sizes.Motivated by Wang et al. [20], we propose operating  1 and  2 by the 2-SWPRT.Specifically, the stopping and decision rules of   ( = 1, 2) by the 2-SWPRT are where where   and   ( = 1, 2) are the boundary parameters (0 <   ,   < ∞).Hence, the stopping and decision rules of the combined 2-SWPRT are defined as 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.
Third, we show that any positive moment of the stopping time is asymptotically optimal on the indifference-zones.
It is clear that the combined 2-SWPRTs have the smallest WESS() in all cases.In fact, compared with the Sobel-Wald and Whitehead-Brunier tests, the WESS() 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(), the combined 2-SWPRT also performs best overall.
From Figure 1, 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  *  ( = 1, 2) and almost the same as the Sobel-Wald test when the true parameter is close to  2−1 or  2 ( = 1, 2).When the true parameter belongs to Θ  ( = 1, 2, 3), the combined 2-SWPRT performs better than the Whitehead-Brunier test and is comparable with the Sobel-Wald test.
Through another simulation study with 10 5 replications, the WESS() and RMI() are presented in Table 2. Similarly, the expected sample sizes for  ∈ (0, 1) are illustrated in Figure 2.
It can be seen from Table 2 that the combined 2-SWPRT still has the smallest WESS() and RMI() for the Bernoulli distribution.Meanwhile, from Figure 2, we have similar

Summary
In this paper, we propose the WESS() to evaluate the overall performance on the indifference-zones for three composite hypotheses' testing problem.In order to minimize WESS() 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.
Similarly, if  >  4 we prefer to accept  3 , and we prefer to accept  2 if  2 <  <  3 .However, we have no strong preference between  1 and  2 if  ∈ [ 1 ,  2 ], and we also have no strong preference between  2 and  3 if  ∈ [ 3 ,  4 ].In these cases, we need more observations for decision.Thus, when the error probabilities satisfy Υ(, ), we focus on reduction of the expected sample sizes over the indifference-zones Θ in applications.Let () be a nonnegative weight function which is sectionally continuous on [ 1 ,  2 ] and [ 3 ,  4 ], respectively, and satisfies ∫ Θ () = 1.We define the weighted expected sample size as For testing problem (2), if  <  1 we prefer to accept  1 and this preference is the stronger the smaller .