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

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

2008-07-16

arxiv.org/abs/0807.2547v2

Electronic Journal of Statistics 2008, Vol. 2, 1345-1372

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/08-EJS267 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by t

Scientific paper

10.1214/08-EJS267

Let $Y$ be a Gaussian vector of $\mathbb{R}^n$ of mean $s$ and diagonal covariance 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 of two 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 $\gamma=1$ corresponds to the homoscedastic case where the components of $Y$ are assumed to have common (unknown) variance. In the opposite, the choice $\gamma>1$ 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 $\{S_m\times\Sigma_m, m\in\mathcal{M}\}$ of parameter 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 of selecting some $\hat{m}$ among $\mathcal{M}$ in such a way that the Kullback risk of $(\hat{s}_{\hat{m}},\hat{\sigma}_{\hat{m}})$ is as close as possible to the minimum of the Kullback risks among the family of estimators $\{(\hat{s}_m,\hat{\sigma}_m), m\in\mathcal{M}\}$. Then we derive uniform rates of 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 to illustrate the performances of our estimators in practice.

Affiliated with

Also associated with

No associations

Rating

Simultaneous estimation of the mean and the variance in heteroscedastic Gaussian regression does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Simultaneous estimation of the mean and the variance in heteroscedastic Gaussian regression, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Simultaneous estimation of the mean and the variance in heteroscedastic Gaussian regression will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-660614