  Let $Y$ be a Gaussian vector whose components are independent with a common
unknown variance. We consider the problem of estimating the mean $\mu$ of $Y$
by model selection. More precisely, we start with a collection
$\mathcal{S}=\{S_m,m\in\mathcal{M}\}$ of linear subspaces of $\mathbb{R}^n$ and
associate to each of these the least-squares estimator of $\mu$ on $S_m$. Then,
we use a data driven penalized criterion in order to select one estimator among
these. Our first objective is to analyze the performance of estimators
associated to classical criteria such as FPE, AIC, BIC and AMDL. Our second
objective is to propose better penalties that are versatile enough to take into
account both the complexity of the collection $\mathcal{S}$ and the sample
size. Then we apply those to solve various statistical problems such as
variable selection, change point detections and signal estimation among others.
Our results are based on a nonasymptotic risk bound with respect to the
Euclidean loss for the selected estimator. Some analogous results are also
established for the Kullback loss.
