  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.
