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In this article, we consider the problem of simultaneous testing ofhypotheses when the individual test statistics are not necessarily independent.Specifically, we consider the problem of simultaneous testing of point nullhypotheses against two-sided alternatives for the mean parameters of normallydistributed random variables. We assume that conditionally given the vector ofmeans, these random variables jointly follow a multivariate normal distributionwith a known but arbitrary covariance matrix. We consider a Bayesian frameworkwhere each unknown mean parameter is modeled through a two-component "spike andslab" mixture prior. This way, unconditionally the test statistics jointly havea mixture of multivariate normal distributions. A new testing procedure isdeveloped that uses the dependence among the test statistics and works in a"step-down" manner. This procedure is general enough to be applied fornon-normal data. A decision theoretic justification in favor of the proposedtesting procedure has been provided by showing that unlike many traditionalp-value based stepwise procedures, this new method possesses a certain"convexity property" which makes it admissible with respect to a vector riskfunction that captures the risks for the individual testing problems. Analternative representation of the proposed test statistics has also beenestablished resulting in great simplification in the computational complexity.It is demonstrated through extensive simulations that for various forms ofdependence and a wide range of sparsity levels, the proposed testing procedurecompares quite favorably with several existing multiple testing proceduresavailable in the literature in terms of overall misclassification probability.

A New Step-down Procedure for Simultaneous Hypothesis Testing Under Dependence