  We prove lower bounds on the error of any estimator for the mean of a real
probability distribution under the knowledge that the distribution belongs to a
given set. We apply these lower bounds both to parametric and nonparametric
estimation. In the nonparametric case, we apply our results to the question of
sub-Gaussian estimation for distributions with finite variance to obtain new
lower bounds in the small error probability regime, and present an optimal
estimator in that regime. In the (semi-)parametric case, we use the Fisher
information to provide distribution-dependent lower bounds that are
constant-tight asymptotically, of order $\sqrt{2\log(1/\delta)/(nI)}$ where $I$
is the Fisher information of the distribution. We use known minimizers of the
Fisher information on some nonparametric set of distributions to give lower
bounds in cases such as corrupted distributions, or bounded/semi-bounded
distributions.
