Open Access
2008 Simultaneous estimation of the mean and the variance in heteroscedastic Gaussian regression
Xavier Gendre
Electron. J. Statist. 2: 1345-1372 (2008). DOI: 10.1214/08-EJS267


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} of parameter sets where Sm and Σm are linear spaces. To each mℳ, 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 among ℳ in such a way that the Kullback risk of (ŝ,σ̂) is as close as possible to the minimum of the Kullback risks among the family of estimators {(ŝm,σ̂m), m}. Then we derive uniform rates of convergence for the estimator (ŝ,σ̂) over Hölderian balls. Finally, we carry out a simulation study in order to illustrate the performances of our estimators in practice.


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Xavier Gendre. "Simultaneous estimation of the mean and the variance in heteroscedastic Gaussian regression." Electron. J. Statist. 2 1345 - 1372, 2008.


Published: 2008
First available in Project Euclid: 29 December 2008

zbMATH: 1320.62092
MathSciNet: MR2471290
Digital Object Identifier: 10.1214/08-EJS267

Primary: 62G08

Keywords: convergence rate , Gaussian regression , Heteroscedasticity , Kullback risk , Model selection

Rights: Copyright © 2008 The Institute of Mathematical Statistics and the Bernoulli Society

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