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

• Published in: Annals of the Institute of Statistical Mathematics 2022-04, Vol.74 (2), p.289-319
• Main Authors: Hoang, Anh-Tuan, Dickhaus, Thorsten
• Format: Article
• Language: English
• Subjects: Economics; Finance; Hypotheses; Insurance; Management; Mathematics; Mathematics and Statistics; Null hypothesis; Parameters; Statistics; Statistics for Business
• ISSN: 0020-3157; 1572-9052
• DOI: 10.1007/s10463-021-00797-0

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