Some Permutationllay Symmetric Multiple Hypotheses Testing Rules Under Dependent Set up

Abstract: In this paper, our interest is in the problem of simultaneous hypothesis testing when the test statistics corresponding to the individual hypotheses are possibly correlated. Specifically, we consider the case when the test statistics together have a multivariate normal distribution (with equal correlation between each pair) with an unknown mean vector and our goal is to decide which components of the mean vector are zero and which are non-zero. This problem was taken up earlier in Bogdan et al. (2011) for the case when the test statistics are independent normals. Asymptotic optimality in a Bayesian decision theoretic sense was studied in this context, the optimal precodures were characterized and optimality of some well-known procedures were thereby established. The case under dependence was left as a challenging open problem. We have studied the problem both theoretically and through extensive simulations and have given some permutation invariant rules. Though in Bogdan et al. (2011), the asymptotic derivations were done in the context of sparsity of the non-zero means, our result does not require the assumption of sparsity and holds under a more general setup.