Abstract

We propose a frequentist testing procedure that maintains a defined coverage and is optimal in the sense that it gives maximal power to detect deviations from a null hypothesis when the alternative to the null hypothesis is sampled from a pre-specified distribution (the prior distribution). Selecting a prior distribution allows to tune the decision rule. This leads to an increased power, if the true data generating distribution happens to be compatible with the prior. It comes at the cost of losing power, if the data generating distribution or the observed data are incompatible with the prior. We illustrate the proposed approach for a binomial experiment, which is sufficiently simple such that the decision sets can be illustrated in figures, which should facilitate an intuitive understanding. The potential beyond the simple example will be discussed: the approach is generic in that the test is defined based on the likelihood function and the prior only. It is comparatively simple to implement and efficient to execute, since it does not rely on Minimax optimization. Conceptually it is interesting to note that for constructing the testing procedure the Bayesian posterior probability distribution is used.

摘要

我们提出了一种常客测试程序，该程序保持定义的覆盖范围，并且在某种意义上是最优的，即当从预先指定的分布（先验分布）中抽取原假设的替代方案时，它提供了最大的能力来检测与原假设的偏差。选择先验分布允许调整决策规则。如果真实的数据生成分布恰好与先验兼容，这将导致功率增加。如果数据生成分布或观察到的数据与先验不兼容，则会以失去功率为代价。我们说明了所提出的二项式实验方法，该方法非常简单，以至于决策集可以用图形来说明，这应该有助于直观理解。将讨论超越简单示例的潜力：该方法是通用的，因为测试是基于似然函数和仅先验定义的。它实现起来相对简单，执行起来也比较高效，因为它不依赖于 Minimax 优化。从概念上讲，有趣的是，为了构建测试程序，使用了贝叶斯后验概率分布。