Pitman Efficiencies of Sequential Tests and Uniform Limit Theorems in Nonparametric Statistics
序贯检验的 Pitman 效率与非参数统计的一致极限定理AI 解读 ↓
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序贯检验的 Pitman 效率与非参数统计的一致极限定理
- 研究问题
- 如何把固定样本检验的 Pitman 效率比较推广到序贯检验，并保证零假设与局部备择下的期望样本量有限？
- 方法与思路
- 作者以标准化统计过程的弱收敛替代渐近正态性，建立一致不变性原理和大偏差定理，并分析两样本位置问题。
- 材料陈述的贡献
- 摘要称，所研究序贯秩检验相对序贯 t 检验的渐近效率，与对应固定样本检验的 Pitman 效率一致。
- 阅读关注点
- 应核对统计量类别、局部备择与停止规则，以及一致极限定理如何同时支持效率比较和期望样本量有限性。
- 阅读边界
- 仅据版本未注明的 OpenAlex 摘要；效率相等的定理条件与具体数值未说明，不能推广到任意序贯检验或备择。
原文摘要
In this paper Pitman's method of constructing and comparing tests based on statistics which are asymptotically normal under the null hypothesis and the local alternatives is extended to sequential tests of statistical hypotheses. The asymptotic normality assumption in Pitman's theory is replaced in its sequential analogue by the weak convergence of normalized processes formed from these statistics under the null hypothesis and the local alternatives. Uniform invariance principles are developed for a large class of statistics, and as an immediate corollary of these results, the desired weak convergence assumption is shown to hold. Furthermore uniform large deviation theorems are obtained for the test statistics and these results guarantee that the sequential tests under consideration have finite expected sample sizes under the null hypothesis and the local alternatives. As an illustration of the general method, the two-sample location problem is studied in detail, and the asymptotic relative efficiencies of the sequential Wilcoxon test, the sequential van der Waerden test and the sequential normal scores test relative to the two-sample sequential $t$-test are easily obtained since one of our key results (Theorem 1) implies that the asymptotic relative efficiencies of these sequential tests coincide with the corresponding Pitman efficiencies of their nonsequential analogues.