A new multiple testing procedure under dependence

In this article, we consider the problem of simultaneous testing of hypotheses when the individual test statistics are not necessarily independent. Specifically, we consider the problem of simultaneous testing of point null hypotheses against two-sided alternatives about the mean parameters of normally distributed random variables. We assume that conditionally given the vector means, these random variables jointly follow a multivariate normal distribution with a known but arbitrary covariance matrix. We consider a Bayesian framework where each unknown mean is modeled via a two component point mass mixture prior, whereby unconditionally the test statistics jointly have a mixture of multivariate normal distributions. A new testing procedure is developed that uses the dependence among the test statistics and works in a step down like manner. This procedure is general enough to be applied to even for non-normal data. A decision theoretic justification in favor of the proposed testing procedure has been provided by showing that unlike the traditional $p$-value based stepwise procedures, this new method possesses a certain convexity property which is essential for the admissibility of a multiple testing procedure with respect to the vector risk function. Consistent estimation of the unknown proportion of alternative hypotheses and variance of the distribution of the non-zero means is theoretically investigated. An alternative representation of the proposed test statistics has also been established resulting in a great reduction in computational complexity. It is demonstrated through extensive simulations that for various forms of dependence and a wide range of sparsity levels, the proposed testing procedure compares quite favourably with several existing multiple testing procedures available in the literature in terms of overall misclassification probability.