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

Abstract

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 σ_i=Γ_i,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 max _iσ_i/min _iσ_i. 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 {S_m×Σ_m, m∈ℳ} of parameter sets where S_m and Σ_m are linear spaces. To each m∈ℳ, we associate a pair of estimators (ŝ_m,σ̂_m) of (s,σ) with values in S_m×Σ_m. Then we design a model selection procedure in view of selecting some m̂ among ℳ in such a way that the Kullback risk of (ŝ_m̂,σ̂_m̂) is as close as possible to the minimum of the Kullback risks among the family of estimators {(ŝ_m,σ̂_m), m∈ℳ}. Then we derive uniform rates of convergence for the estimator (ŝ_m̂,σ̂_m̂) over Hölderian balls. Finally, we carry out a simulation study in order to illustrate the performances of our estimators in practice.