On the usage of randomized p-values in the Schweder-Spjotvoll estimator

We consider multiple test problems with composite null hypotheses and the estimation of the proportion pi(0) of true null hypotheses. The Schweder-Spjotvoll estimator pi(0) utilizes marginal p-values and relies on the assumption that p-values corresponding to true nulls are uniformly distributed on [0, 1]. In the case of composite null hypotheses, marginal p-values are usually computed under least favorable parameter configurations (LFCs). Thus, they are stochastically larger than uniform under non-LFCs in the null hypotheses. When using these LFC-based p-values, pi(0) tends to overestimate pi(0). We introduce a new way of randomizing p-values that depends on a tuning parameter c is an element of [0, 1]. For a certain value c = c*, the resulting bias of pi(0) is minimized. This often also entails a smaller mean squared error of the estimator as compared to the usage of LFC-based p-values. We analyze these points theoretically, and we demonstrate them numerically in simulations.

Automated summary

This study examines multiple test problems with composite null hypotheses, introducing the Schweder-Spjotvoll estimator and the limitations related to relying on least favorable parameter configurations. A new method of randomizing p-values based on a tuning parameter is presented, showing reduced bias and mean squared error compared to using LFC-based p-values. Theoretical analysis and numerical simulations demonstrate the effectiveness of this approach.