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June 2016 Supremum norm posterior contraction and credible sets for nonparametric multivariate regression
William Weimin Yoo, Subhashis Ghosal
Ann. Statist. 44(3): 1069-1102 (June 2016). DOI: 10.1214/15-AOS1398


In the setting of nonparametric multivariate regression with unknown error variance $\sigma^{2}$, we study asymptotic properties of a Bayesian method for estimating a regression function $f$ and its mixed partial derivatives. We use a random series of tensor product of B-splines with normal basis coefficients as a prior for $f$, and $\sigma$ is either estimated using the empirical Bayes approach or is endowed with a suitable prior in a hierarchical Bayes approach. We establish pointwise, $L_{2}$ and $L_{\infty}$-posterior contraction rates for $f$ and its mixed partial derivatives, and show that they coincide with the minimax rates. Our results cover even the anisotropic situation, where the true regression function may have different smoothness in different directions. Using the convergence bounds, we show that pointwise, $L_{2}$ and $L_{\infty}$-credible sets for $f$ and its mixed partial derivatives have guaranteed frequentist coverage with optimal size. New results on tensor products of B-splines are also obtained in the course.


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William Weimin Yoo. Subhashis Ghosal. "Supremum norm posterior contraction and credible sets for nonparametric multivariate regression." Ann. Statist. 44 (3) 1069 - 1102, June 2016.


Received: 1 November 2014; Revised: 1 September 2015; Published: June 2016
First available in Project Euclid: 11 April 2016

zbMATH: 1338.62121
MathSciNet: MR3485954
Digital Object Identifier: 10.1214/15-AOS1398

Primary: 62G08
Secondary: 62G05 , 62G15 , 62G20

Keywords: anisotropic smoothness , mixed partial derivatives , nonparametric multivariate regression , sup-norm posterior contraction , Tensor product B-splines

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 3 • June 2016
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