SUPREMUM NORM POSTERIOR CONTRACTION AND CREDIBLE SETS FOR NONPARAMETRIC MULTIVARIATE REGRESSION

• 成果类型: Article
• 署名作者: Yoo, William Weimin; Ghosal, Subhashis
• 署名单位: Universite PSL; Universite Paris-Dauphine; North Carolina State University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/15-AOS1398
• 发表日期: 2016
• 页码: 1069-1102
• 关键词: bernstein-von-mises; gaussian white-noise; confidence bands; Adaptive estimation; DENSITY-ESTIMATION; rates; priors; CONVERGENCE; METRICS

摘要

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-infinity-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-infinity-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.