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Bernstein-von Mises theorems for Gaussian regression with increasing number of regressors. (English) Zbl 1231.62061
Summary: 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.

62G08 Nonparametric regression and quantile regression
62F15 Bayesian inference
62J05 Linear regression; mixed models
62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation
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