Stepwise Likelihood Ratio Statistics in Sequential Studies
序贯研究中的逐步似然比统计量AI 解读 ↓
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序贯研究中的逐步似然比统计量
- 研究问题
- 当备择假设含未知参数时，如何构造可持续更新的似然比统计量，使序贯检验仍具有跨所有观察时点的越界概率控制？
- 方法与思路
- 作者用此前观测估计备择参数，将每个新观测的条件似然比累乘，并以正态和指数响应的模拟比较固定边界检验。
- 材料陈述的贡献
- 摘要称零假设下统计量曾超过正阈值c的概率不超过1/c，而复合备择下通常依概率趋于无穷，支持其序贯检验用途。
- 阅读关注点
- 应核对条件似然构造、参数估计的可测性与正则条件，以及备择下发散的适用范围和模拟中的错误率与停止时间。
- 阅读边界
- 摘要未说明全部模型条件，也未报告模拟比较的具体表现；不能推断该统计量在所有复合备择下发散或优于已有方法。
原文摘要
Summary It is well known that in a sequential study the probability that the likelihood ratio for a simple alternative hypothesis H1versus a simple null hypothesis H0 will ever be greater than a positive constant c will not exceed 1/c under H0. However, for a composite alternative hypothesis, this bound of 1/c will no longer hold when a generalized likelihood ratio statistic is used. We consider a stepwise likelihood ratio statistic which, for each new observation, is updated by cumulatively multiplying the ratio of the conditional likelihoods for the composite alternative hypothesis evaluated at an estimate of the parameter obtained from the preceding observations versus the simple null hypothesis. We show that, under the null hypothesis, the probability that this stepwise likelihood ratio will ever be greater than c will not exceed 1/c. In contrast, under the composite alternative hypothesis, this ratio will generally converge in probability to ∞. These results suggest that a stepwise likelihood ratio statistic can be useful in a sequential study for testing a composite alternative versus a simple null hypothesis. For illustration, we conduct two simulation studies, one for a normal response and one for an exponential response, to compare the performance of a sequential test based on a stepwise likelihood ratio statistic with a constant boundary versus some existing approaches.