Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors

Details Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors

2010-09-07

arxiv.org/abs/1009.1370v3

Annals of Statistics 39, 5 (2011) 2557-2584

Mathematics

Statistics Theory

Scientific paper

10.1214/11-AOS912

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.

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