The idea of an optimal test statistic in the context of simultaneous hypothesis testing was given by Sun and Tony Cai (2009) which is the conditional probability of a hypothesis being null given the data. Since we do not have a simplified expression of the statistic, it is impossible to implement the optimal test in more general dependency setup. This note simplifies the expression of optimal test statistic of Sun and Tony Cai (2009) under the multivariate normal model. We have considered the model of Xie et. al.(2011), where the test statistics are generated from a multivariate normal distribution conditional to the unobserved states of the hypotheses and the states are i.i.d. Bernoulli random variables. While the equivalence of LFDR and optimal test statistic was established under very stringent conditions of Xie et. al.(2016), the expression obtained in this paper is valid for any covariance matrix and for any fixed 0<p<1. The optimal procedure is implemented with the help of this expression and the performances have been compared with Benjamini Hochberg method and marginal procedure.

翻译：在多重假设检验背景下，最优检验统计量的概念由Sun与Tony Cai (2009)提出，该统计量定义为给定数据时原假设成立的条件概率。由于该统计量缺乏简化表达式，导致其在更一般的相依结构下无法实现。本文简化了多元正态模型下Sun与Tony Cai (2009)最优检验统计量的表达式。我们采用Xie等(2011)的模型设定，其中检验统计量来源于给定假设未观测状态条件下的多元正态分布，且状态为独立同分布的伯努利随机变量。虽然Xie等(2016)已建立局部错误发现率与最优检验统计量在极严格条件下的等价性，但本文获得的表达式对任意协方差矩阵及任意固定参数0<p<1均成立。基于该表达式实现最优检验程序，并将其性能与Benjamini-Hochberg方法及边缘检验程序进行对比。