  In this article, we investigate certain asymptotic optimality properties of a
very broad class of one-group continuous shrinkage priors for simultaneous
estimation and testing of a sparse normal mean vector. Asymptotic optimality of
Bayes estimates and posterior concentration properties corresponding to the
general class of one-group priors under consideration are studied where the
data is assumed to be generated according to a multivariate normal distribution
with a fixed unknown mean vector. Under the assumption that the number of
non-zero means is known, we show that Bayes estimators arising out of this
general class of shrinkage priors under study, attain the minimax risk, up to
some multiplicative constant, under the $l_2$ norm. In particular, it is shown
that for the horseshoe-type priors such as the three parameter beta normal
mixtures with parameters $a=0.5, b>0$ and the generalized double Pareto prior
with shape parameter $\alpha=1$, the corresponding Bayes estimates become
asymptotically minimax. Moreover, posterior distributions arising out of this
general class of one-group priors are shown to contract around the true mean
vector at the minimax $l_2$ rate for a wide range of values of the global
shrinkage parameter depending on the proportion of non-zero components of the
underlying mean vector. An important and remarkable fact that emerges as a
consequence of one key result essential for proving the aforesaid minimaxity
result is that, within the asymptotic framework of Bogdan et al. (2011), the
natural thresholding rules due to Carvalho et al. (2010) based on the
horseshoe-type priors, asymptotically attain the optimal Bayes risk w.r.t. a
$0-1$ loss, up to the correct multiplicative constant and are thus,
asymptotically Bayes optimal under sparsity (ABOS).
