文章基本信息

标题：Sparse model selection under heterogeneous noise: Exact penalisation and data-driven thresholding
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作者：Laurent Cavalier ; Markus Reiß
期刊名称：Electronic Journal of Statistics
印刷版ISSN：1935-7524
出版年度：2014
卷号：8
期号：1
页码：432-455
DOI：10.1214/14-EJS889
语种：English
出版社：Institute of Mathematical Statistics
摘要：We consider a Gaussian sequence space model $X_{\lambda}=f_{\lambda}+\xi_{\lambda},$ where the noise variables $(\xi_{\lambda})_{\lambda}$ are independent, but with heterogeneous variances $(\sigma_{\lambda}^{2})_{\lambda}$. Our goal is to estimate the unknown signal vector $(f_{\lambda})$ by a model selection approach. We focus on the situation where the non-zero entries $f_{\lambda}$ are sparse. Then the heterogenous case is much more involved than the homogeneous model where $\sigma_{\lambda}^{2}=\sigma^{2}$ is constant. Indeed, we can no longer profit from symmetry inside the stochastic process that one needs to control. The problem and the penalty do not only depend on the number of coefficients that one selects, but also on their position. This appears also in the minimax bounds where the worst coefficients will go to the larger variances. With a careful and explicit choice of the penalty, however, we are able to select the correct coefficients and get a sharp non-asymptotic control of the risk of our procedure. Some finite sample results from simulations are provided.