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The idea of an optimal test statistic in the context of simultaneoushypothesis testing was given by Sun and Tony Cai (2009) which is theconditional probability of a hypothesis being null given the data. Since we donot have a simplified expression of the statistic, it is impossible toimplement the optimal test in more general dependency setup. This notesimplifies 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 normaldistribution conditional to the unobserved states of the hypotheses and thestates are i.i.d. Bernoulli random variables. While the equivalence of LFDR andoptimal test statistic was established under very stringent conditions of Xieet. al.(2016), the expression obtained in this paper is valid for anycovariance matrix and for any fixed 0<p<1. The optimal procedure is implementedwith the help of this expression and the performances have been compared withBenjamini Hochberg method and marginal procedure.

Optimal test statistic under normality assumption