论文标题
关于在Schweder-Spjotvoll估算器中使用随机p值的使用
On the usage of randomized p-values in the Schweder-Spjotvoll estimator
论文作者
论文摘要
我们关注的是复合零假设的多个测试问题，以及对真实零假设的比例$π_{0} $的估计。 Schweder-spjøtvoll估算器$ \hatπ_0$使用边缘$ p $ - 值，只有当与真实的null Null假设相对应的$ p $ - 价值时，才能正常工作。在复合零假设的情况下，边际$ p $值通常是在最不利的参数配置（LFC）下计算的。因此，在零假设中的非LFC下，它们的随机性大于$ \ mathrm {uni} [0,1] $。当使用这些基于LFC的$ p $ - 值时，$ \hatπ_0$倾向于高估$π_{0} $。我们介绍了一种随机化$ p $ - 值的新方法，该方法取决于[0,1] $中的调谐参数$ c \，以便$ c = 0 $ and $ c = 1 $导致$ \ mathrm {uni} [0,1] $ - 分布$ p $ - values，这些$ p $ values与原始lfc $ p $ p $ p $ p $ p $ - valuese无关。对于某个值，$ c = c = c^{\ star} $当使用我们的随机$ p $ - 价值时，$ \hatπ_0$的偏差将最小化。与基于LFC的$ p $值相比，这通常还需要估算器的均值较小。我们从理论上分析了这些点，并在各种标准统计模型下在计算机模拟中以数字为单位进行了分析。
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-Spjøtvoll estimator $\hatπ_0$ utilizes marginal $p$-values and only works properly if the $p$-values that correspond to the true null hypotheses are uniformly distributed on $[0,1]$ ($\mathrm{Uni}[0,1]$-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 $\mathrm{Uni}[0,1]$ under non-LFCs in the null hypotheses. When using these LFC-based $p$-values, $\hatπ_0$ tends to overestimate $π_{0}$. We introduce a new way of randomizing $p$-values that depends on a tuning parameter $c\in[0,1]$, such that $c=0$ and $c=1$ lead to $\mathrm{Uni}[0,1]$-distributed $p$-values, which are independent of the data, and to the original LFC-based $p$-values, respectively. For a certain value $c=c^{\star}$ the bias of $\hatπ_0$ 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.
