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：本文重新审视了多元正态均值双边同时检验中的以下开放问题：Benjamini和Hochberg（BH，1995）的方法能否在不施加任何相关性依赖结构的情况下控制错误发现率（FDR）？该问题的答案通常被认为是肯定的，且文献中基于大量数值研究报告的结果强烈支持这一猜想。据我们所知，目前尚无理论证明证实这一答案。本文对该猜想提供了部分证明。具体而言，我们考虑以下两种情形：（i）协方差矩阵已知；（ii）协方差矩阵为已知矩阵的未知标量倍数。我们证明，在这两种情形下，基于原始z检验或t检验统计量的加权版本的BH型逐步上升方法均可控制FDR。