Bayes oracle property of multiple tests of multivariate normal means under sparsity

The paper considers a multiple testing problem of multivariate normal means under sparsity. First, the Bayes risk of the multivariate Bayes oracle is derived. Then, a hierarchical Bayesian approach is taken with global–local shrinkage priors, where the global parameter is either treated as a tuning parameter or is given a specific prior. The method is shown to attain an asymptotic Bayes optimal under sparsity (ABOS) property. Finally, an empirical Bayes procedure is proposed which involves estimation of the global shrinkage parameter. The approach is also shown to lead to the ABOS property.