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Semiparametric Bernstein-von Mises for the error standard deviation. (English) Zbl 1337.62087
Summary: We study Bayes procedures for nonparametric regression problems with Gaussian errors, giving conditions under which a Bernstein-von Mises result holds for the marginal posterior distribution of the error standard deviation. We apply our general results to show that a single Bayes procedure using a hierarchical spline-based prior on the regression function and an independent prior on the error variance, can simultaneously achieve adaptive, rate-optimal estimation of a smooth, multivariate regression function and efficient, \(\sqrt{n}\)-consistent estimation of the error standard deviation.

MSC:
62G08 Nonparametric regression and quantile regression
62G09 Nonparametric statistical resampling methods
62F15 Bayesian inference
62C10 Bayesian problems; characterization of Bayes procedures
62G20 Asymptotic properties of nonparametric inference
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