In this article, we consider confidence set estimation of the mean vector of a p-dimensional normal distribution, based on n iid observations from it, with the covariance matrix \(\Sigma \) assumed positive definite but completely unknown. We consider two regimes, p fixed and p growing with n. The coverage properties as well as the expected volume and the girth of the classic Hotelling confidence set and Bayesian HPD sets under normal priors are worked out in a series of theorems. Of specific interest are theorems on the existence of an explicit phase transition in the coverage probability of the Hotelling confidence set and subexponential bounds on the noncoverage probability of HPD sets when the mean vector is sparse in \(\mathcal {L}_0\) and \(\mathcal {L}_1\) norms. The results are illustrated with discussions and two sets of computations.
