This paper revisits the following open question in simultaneous testing of multivariate normal means against two-sided alternatives: Can the method of Benjamini and Hochberg (BH, 1995) control the false discovery rate (FDR) without imposing any dependence structure on the correlations? The answer to this question is generally believed to be yes, and is conjectured so in the literature since results of numerical studies investigating the question and reported in numerous papers strongly support it. No theoretical justification of this answer has yet been put forward in the literature, as far as we know. In this paper, we offer a partial proof of this conjecture. More specifically, we consider the following two settings - (i) the covariance matrix is known and (ii) the covariance matrix is an unknown scalar multiple of a known matrix - and prove that in each of these settings a BH-type stepup method based on some weighted versions of the original z- or t-test statistics controls the FDR.

翻译：本文重新审视多元正态分布均值的同时检验中以下开放问题：本杰明和霍奇伯格方法（BH，1995）能否在不强加相关性结构的情况下控制假阳性发现率（FDR）？一般认为答案是肯定的，因为数值研究的结果证明了这一点，并且在许多论文中也被猜测。据我们所知，还没有在文献中提供这个答案的理论证明。在本文中，我们提供了这个猜想的部分证明。更具体地说，我们考虑以下两个情况——（i）协方差矩阵已知，和（ii）协方差矩阵是已知矩阵的未知标量倍数——并且证明在这些情况下，基于一些加权版本的原始z-或t检验统计量的BH类型阶段上方法可以控制FDR。