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

我们考虑了带有复合零假设和真实零假设的比例\（\ pi _ {0} \）的多个检验问题。Schweder–Spjøtvoll估计量\（{\ hat {\ pi}} _ 0 \）利用边际p值，并假设与真实null对应的p值均匀分布在[0，1]上。在复合零假设的情况下，通常在最不利的参数配置（LFC）下计算边际p值。因此，在原假设中，它们在非LFC下随机大于均匀。使用这些基于LFC的p值时，\（{\ hat {\ pi}} _ 0 \）往往会高估\（\ pi _ {0} \）。我们介绍了一种根据调整参数\（c \ in [0,1] \）随机化p值的新方法。对于某个值\（c = c ^ {\ star} \），\（{\ hat {\ pi}} _ 0 \）的最终偏差被最小化。与基于LFC的p值的使用相比，这通常还需要估计器的均方误差较小。我们从理论上分析这些点，并在仿真中以数值方式对其进行演示。