  We consider the problem of estimating the mean $f$ of a Gaussian vector $Y$
with independent components of common unknown variance $\sigma^{2}$. Our
estimation procedure is based on estimator selection. More precisely, we start
with an arbitrary and possibly infinite collection $\FF$ of estimators of $f$
based on $Y$ and, with the same data $Y$, aim at selecting an estimator among
$\FF$ with the smallest Euclidean risk. No assumptions on the estimators are
made and their dependencies with respect to $Y$ may be unknown. We establish a
non-asymptotic risk bound for the selected estimator. As particular cases, our
approach allows to handle the problems of aggregation and model selection as
well as those of choosing a window and a kernel for estimating a regression
function, or tuning the parameter involved in a penalized criterion. We also
derive oracle-type inequalities when $\FF$ consists of linear estimators. For
illustration, we carry out two simulation studies. One aims at comparing our
procedure to cross-validation for choosing a tuning parameter. The other shows
how to implement our approach to solve the problem of variable selection in
practice.
