BERNSTEIN-VON MISES THEOREMS FOR GAUSSIAN REGRESSION WITH INCREASING NUMBER OF REGRESSORS

• 成果类型: Article
• 署名作者: Bontemps, Dominique
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris Saclay
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/11-AOS912
• 发表日期: 2011
• 页码: 2557-2584
• 关键词: posterior distributions; Asymptotic Normality; exponential-families; convergence-rates

摘要

This paper brings a contribution to the Bayesian theory of nonparametric and semiparametric estimation. We are interested in the asymptotic normality of the posterior distribution in Gaussian linear regression models when the number of regressors increases with the sample size. Two kinds of Bernstein-von Mises theorems are obtained in this framework: nonparametric theorems for the parameter itself, and semiparametric theorems for functionals of the parameter. We apply them to the Gaussian sequence model and to the regression of functions in Sobolev and C-alpha classes, in which we get the minimax convergence rates. Adaptivity is reached for the Bayesian estimators of functionals in our applications.