Simultaneous estimation of the mean and the variance in heteroscedastic Gaussian regression

26 pagesLet $Y$ be a Gaussian vector of $\R^n$ of mean $s$ and diagonal covariance matrix $\Gamma$. Our aim is to estimate both $s$ and the entries $\sigma_i=\Gamma_{i,i}$, for $i=1,\dots,n$, on the basis of the observation of two independent copies of $Y$. Our approach is free of any prior assumption on $s$ but requires that we know some upper bound $\gamma$ on the ratio $\max_i\sigma_i/\min_i\sigma_i$. Our estimation strategy is based on model selection. We consider a family $\{S_m\times\Sigma_m,\ m\in\mathcal{M}\}$ of parameter sets where $S_m$ and $\Sigma_m$ are linear spaces. To each $m\in\mathcal{M}$, we associate a pair of estimators $(\hat{s}_m,\hat{\sigma}_m)$ of $(s,\sigma)$ with values in $S_m\times\Sigma_m$. Then we design a model selection procedure in view of selecting some $\hat{m}$ among $\mathcal{M}$ in such way that the Kullback risk of $(\hat{s}_{\hat{m}},\hat{\sigma}_{\hat{m}})$ is as close as possible to the minimum of the Kullback risks among the family of estimators $\{(\hat{s}_m,\hat{\sigma}_m),\ m\in\mathcal{M}\}$. Then we derive uniform rates of convergence for the estimator $(\hat{s}_{\hat{m}},\hat{\sigma}_{\hat{m}})$ over Hölderian balls