Asymptotic bayes optimality under sparsity for equicorrelated multivariate normal test statistics

Here we address dependence among the test statistics in connection with asymptotically Bayes' optimal tests in presence of sparse alternatives. Extending the setup in Bogdan et.al. (2011) we consider an equicorrelated ( with equal correlation $\rho$ ) multivariate normal assumption on the joint distribution of the test statistics, while conditioned on the mean vector $\boldsymbol{\mu}$. Rest of the set up is identical to Bogdan et.al. (2011) with a slight modification in the asymptotic framework. We exploit an well known result on equicorrelated multivariate normal variables with equal marginal variances to decompose the test statistics into independent random variables. We then identify a set of independent yet unobservable gaussian random variables sufficient for the multiple testing problem and chalk out the necessary and sufficient conditions for single cutoff tests to be ABOS based on those dummy variables following Bogdan et.al. (2011). Further we replaced the dummy variables with deviations of the statistics from their arithmetic means which were easily calculable from the observations due to the decomposition used earlier. Additional assumptions are then derived so that the necessary and sufficient conditions for single cutoff tests to be ABOS using the independent dummy variables plays the same role with the replacement variable as well (with a deviation of order $o(1)$). Next with the same additional assumption, necessary and sufficient conditions for single cutoff tests to control the Bayesian FDRs are derived and as a consequence under various sparsity assumptions we proved that the classical Bonferroni and Benjamini-Hochberg methods of multiple testing are ABOS if the same conditions are satisfied.