On the usage of randomized p-values in the Schweder-Spjotvoll estimator
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We are concerned with multiple test problems with composite null hypotheses and the estimation of the proportion π_0 of true null hypotheses. The Schweder-Spjø tvoll estimator π̂_0 utilizes marginal p-values and only works properly if the p-values that correspond to the true null hypotheses are uniformly distributed on [0,1] (Uni[0,1]-distributed). 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 Uni[0,1] under non-LFCs in the null hypotheses. When using these LFC-based p-values, π̂_0 tends to overestimate π_0. We introduce a new way of randomizing p-values that depends on a tuning parameter c∈[0,1], such that c=0 and c=1 lead to Uni[0,1]-distributed p-values, which are independent of the data, and to the original LFC-based p-values, respectively. For a certain value c=c^ the bias of π̂_0 is minimized when using our randomized p-values. This often also entails a smaller mean squared error of the estimator as compared to the usage of the LFC-based p-values. We analyze these points theoretically, and we demonstrate them numerically in computer simulations under various standard statistical models.
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