  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.
