  Consider the problem of simultaneous testing for the means of independent
normal observations. In this paper, we study some asymptotic optimality
properties of certain multiple testing rules induced by a general class of
one-group shrinkage priors in a Bayesian decision theoretic framework, where
the overall loss is taken as the number of misclassified hypotheses. We assume
a two-groups normal mixture model for the data and consider the asymptotic
framework adopted in Bogdan et al. (2011) who introduced the notion of
asymptotic Bayes optimality under sparsity in the context of multiple testing.
The general class of one-group priors under study is rich enough to include,
among others, the families of three parameter beta, generalized double Pareto
priors, and in particular the horseshoe, the normal-exponential-gamma and the
Strawderman-Berger priors. We establish that within our chosen asymptotic
framework, the multiple testing rules under study asymptotically attain the
risk of the Bayes Oracle up to a multiplicative factor, with the constant in
the risk close to the constant in the Oracle risk. This is similar to a result
obtained in Datta and Ghosh (2013) for the multiple testing rule based on the
horseshoe estimator introduced in Carvalho et al. (2009, 2010). We further show
that under very mild assumption on the underlying sparsity parameter, the
induced decision rules based on an empirical Bayes estimate of the
corresponding global shrinkage parameter proposed by van der Pas et al. (2014),
attain the optimal Bayes risk up to the same multiplicative factor
asymptotically. We provide a unifying argument applicable for the general class
of priors under study. In the process, we settle a conjecture regarding
optimality property of the generalized double Pareto priors made in Datta and
Ghosh (2013). Our work also shows that the result in Datta and Ghosh (2013) can
be improved further.
