Multiple testing procedures for discrete 𝑝-values and composite null hypotheses

This research considers randomized 𝑝-values for testing composite null hypotheses in discrete models. We assume throughout that a real-valued test statistic 𝑇 such that large values indicate that the data is incompatible with the null model is available. We further let this test statistic be discrete, and since the 𝑝-values are a deterministic transformation of these test statistics, the resulting 𝑝-values will also be discrete. Furthermore, we consider distributions with the monotone likelihood ratio (MLR) property in the test statistic 𝑇, for example, any one-dimensional exponential family of distributions. Specifically, Chapter 2 considers the composite null hypothesis problem for a discrete model. As earlier mentioned, the 𝑝-value can fail to meet the uniformity requirement, among others, when dealing with a composite null hypothesis or discrete test statistic. We propose a singlestage randomized 𝑝-value as a remedy. The single-stage randomized 𝑝-value is based on the least favorable parameter configuration (LFC) 𝑝-value and only deals with the conservativeness resulting from the composite nature of our null hypothesis. This randomized 𝑝-value also partially deals with the discreteness of the test statistic. As a further remedy, we propose a two-stage randomized 𝑝-value. Here, we randomize in the first stage to deal with the discrete test statistic. The second stage of randomization deals with the composite nature of our null hypothesis. In Chapter 3,we extend our two-stage randomized 𝑝-value to the case of an interval composite null hypothesis, where the null hypothesis is decomposed into two one-sided hypotheses, leading to two composite null hypotheses. In both Chapters 2 and 3, we illustrate and provide a mathematical proof showing that the two-stage randomized 𝑝-value is strictly increasing with an increase in the sample size. We further illustrate in both chapters that the 𝑝-value is less conservative compared to the other three 𝑝-values (LFC, UMP, and the single-stage randomized 𝑝-values). This conservativeness further reduces with an increase in the sample size. Finally, we give a small-scale simulation study to illustrate that the two-stage randomized 𝑝-value gives the best estimates of the proportion of true null hypotheses in multiple testing when using the Schweder and Spjøtvoll, 1982 estimator.