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This paper revisits the following open question in simultaneous testing ofmultivariate normal means against two-sided alternatives: Can the method ofBenjamini and Hochberg (BH, 1995) control the false discovery rate (FDR)without imposing any dependence structure on the correlations? The answer tothis question is generally believed to be yes, and is conjectured so in theliterature since results of numerical studies investigating the question andreported in numerous papers strongly support it. No theoretical justificationof 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 someweighted versions of the original z- or t-test statistics controls the FDR.

On Controlling the False Discovery Rate in Multiple Testing of the Means of Correlated Normals Against Two-Sided Alternatives