On the usage of randomized p-values in the Schweder–Spjøtvoll estimator

We consider multiple test problems with composite null hypotheses and the estimation of the proportion $$\pi _{0}$$ π 0 of true null hypotheses. The Schweder–Spjøtvoll estimator $${\hat{\pi }}_0$$ π ^ 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, $${\hat{\pi }}_0$$ π ^ 0 tends to overestimate $$\pi _{0}$$ π 0 . We introduce a new way of randomizing p-values that depends on a tuning parameter $$c \in [0,1]$$ c ∈ [ 0 , 1 ] . For a certain value $$c = c^{\star }$$ c = c ⋆ , the resulting bias of $${\hat{\pi }}_0$$ π ^ 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.