Improving Efficiency of Tests for Composite Null Hypotheses

The goal of mediation analysis is to study the effect of exposure on an outcome interceded by a mediator. Two simple hypotheses are tested: the effect of the exposure on the mediator, and the effect of the mediator on the outcome. When either of these hypotheses is true, a predetermined significance level can be assured. When both nulls are true, the same test becomes conservative. Adaptively finding the correct scenario enables customizing the tests and consequently enlarges their efficiency, which is most important in a multiple testing framework. In this work, we link between adaptive two-stage procedures and shrinkage estimators. We first study the properties of shrinkage estimators, and characterize their behavior at different parameter points using local asymptotics. We formulate theoretical results regarding shrinkage estimators, compared to regular estimators. We then discuss the multiple-testing framework and state results about using shrinkage estimator in two-stage procedures for controlling the FWER. Taking advantage of these theoretical results, we suggest a number of estimators and test statistics for the two-stage mediation procedures. We then investigate their empirical FWER and power, compared to regular estimators and tests, through simulations.