  We consider the horseshoe estimator due to Carvalho, Polson and Scott (2010)
for the multivariate normal mean model in the situation that the mean vector is
sparse in the nearly black sense. We assume the frequentist framework where the
data is generated according to a fixed mean vector. We show that if the number
of nonzero parameters of the mean vector is known, the horseshoe estimator
attains the minimax $\ell_2$ risk, possibly up to a multiplicative constant. We
provide conditions under which the horseshoe estimator combined with an
empirical Bayes estimate of the number of nonzero means still yields the
minimax risk. We furthermore prove an upper bound on the rate of contraction of
the posterior distribution around the horseshoe estimator, and a lower bound on
the posterior variance. These bounds indicate that the posterior distribution
of the horseshoe prior may be more informative than that of other one-component
priors, including the Lasso.
