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

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Let Y be a Gaussian vector of Rⁿ of mean s and diagonal covariance matrix Γ. Our aim is to estimate both s and the entries σᵢ=Γᵢ,ᵢ, for i=1,...,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 γ on the ratio ᵢσᵢ/ᵢσᵢ. For example, the choice γ=1 corresponds to the homoscedastic case where the components of Y are assumed to have common (unknown) variance. In the opposite, the choice γ>1 corresponds to the heteroscedastic case where the variances of the components of Y are allowed to vary within some range. Our estimation strategy is based on model selection. We consider a family {Sₘ×Σₘ, m} of parameter sets where Sₘ and Σₘ are linear spaces. To each m, we associate a pair of estimators (sₘ,σₘ) of (s,σ) with values in Sₘ×Σₘ. Then we design a model selection procedure in view of selecting some m among M in such a way that the Kullback risk of (sm,σm) is as close as possible to the minimum of the Kullback risks among the family of estimators {(sₘ,σₘ), m}. Then we derive uniform rates of convergence for the estimator (sm,σm) over Hölderian balls. Finally, we carry out a simulation study in order to illustrate the performances of our estimators in practice.