Given a fixed-sample-size test that controls the error probabili-ties under two specific, but arbitrary, distributions, a 3-stage and two 4-stage tests are proposed and analyzed. For each of them, a novel, concrete, non-asymptotic, non-conservative design is specified, which guarantees the same error control as the given fixed-sample-size test. Moreover, first-order asymptotic approximation are established on their expected sample sizes under the two prescribed distributions as the error probabilities go to zero. As a corollary, it is shown that the proposed multistage tests can achieve, in this asymptotic sense, the optimal expected sample size under these two distributions in the class of all sequential tests with the same error control. Furthermore, they are shown to be much more robust than Wald's SPRT when applied to one-sided testing problems and the error probabilities under control are small enough. These general results are applied to testing problems in the iid setup and beyond, such as testing the correlation coefficient of a first-order autoregression, or the transition matrix of a finite-state Markov chain, and are illustrated in various numerical studies.
翻译:根据一个固定的抽样规模测试,在两种特定但任意的分布、3阶段和2个4阶段的测试下控制出错,因此建议并分析两种特定但任意的分布、3阶段和2个4阶段的测试。对于每一种测试,都指定了一种新型的、混凝土的、非消毒的、非保守性的设计,保证与给定的固定抽样大小的测试一样的错误控制。此外,在两种规定的分布下,其预期的样本大小上都建立了一阶的无症状近似,因为误差概率会降至零。作为必然结果,表明拟议的多阶段测试可以实现,从这种无症状意义上讲,在所有顺序测试的类别中,在这两种分布下的预期最佳样本规模是相同的错误控制。此外,在将沃尔德的SPRT应用于单方测试问题时,它们比沃尔德的SPRT强得多,而且所控制的误差概率也足够小。这些一般结果用于测试iid设置和以后的问题,例如测试第一级自动的自闭式自闭式的自闭式研究、示式数矩阵和各种定式矩阵的过渡。