The Annals of Statistics
- Ann. Statist.
- Volume 39, Number 5 (2011), 2557-2584.
Bernstein–von Mises theorems for Gaussian regression with increasing number of regressors
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 Cα classes, in which we get the minimax convergence rates. Adaptivity is reached for the Bayesian estimators of functionals in our applications.
Ann. Statist., Volume 39, Number 5 (2011), 2557-2584.
First available in Project Euclid: 30 November 2011
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Bontemps, Dominique. Bernstein–von Mises theorems for Gaussian regression with increasing number of regressors. Ann. Statist. 39 (2011), no. 5, 2557--2584. doi:10.1214/11-AOS912. https://projecteuclid.org/euclid.aos/1322663468
- Supplementary material: Supplement to “Bernstein–von Mises theorems for Gaussian regression with increasing number of regressors”. This contains the proofs of various technical results stated in the main article “Bernstein–von Mises Theorems for Gaussian regression with increasing number of regressors.”.