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

Gendre, Xavier Simultaneous estimation of the mean and the variance in heteroscedastic Gaussian regression. (2008) Electronic Journal of Statistics, 2. 1345-1372. ISSN 1935-7524

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Official URL: https://doi.org/10.1214/08-EJS267

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

Item Type:Article
HAL Id:hal-02045587
Audience (journal):International peer-reviewed journal
Uncontrolled Keywords:
Institution:French research institutions > Centre National de la Recherche Scientifique - CNRS (FRANCE)
Other partners > Université Côte d'Azur (FRANCE)
Laboratory name:
Statistics:download
Deposited By: Xavier Gendre
Deposited On:22 Feb 2019 09:54

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