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Let $Y$ be a Gaussian vector of $\mathbb{R}^n$ of mean $s$ and diagonalcovariance matrix $\Gamma$. Our aim is to estimate both $s$ and the entries$\sigma_i=\Gamma_{i,i}$, for $i=1,...,n$, on the basis of the observation oftwo 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$\max_i\sigma_i/\min_i\sigma_i$. For example, the choice $\gamma=1$ correspondsto the homoscedastic case where the components of $Y$ are assumed to havecommon (unknown) variance. In the opposite, the choice $\gamma>1$ correspondsto the heteroscedastic case where the variances of the components of $Y$ areallowed to vary within some range. Our estimation strategy is based on modelselection. We consider a family $\{S_m\times\Sigma_m, m\in\mathcal{M}\}$ ofparameter 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 ofselecting some $\hat{m}$ among $\mathcal{M}$ in such a way that the Kullbackrisk of $(\hat{s}_{\hat{m}},\hat{\sigma}_{\hat{m}})$ is as close as possible tothe minimum of the Kullback risks among the family of estimators$\{(\hat{s}_m,\hat{\sigma}_m), m\in\mathcal{M}\}$. Then we derive uniform ratesof convergence for the estimator $(\hat{s}_{\hat{m}},\hat{\sigma}_{\hat{m}})$over H\"{o}lderian balls. Finally, we carry out a simulation study in order toillustrate the performances of our estimators in practice.

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