COVERAGE OF CREDIBLE INTERVALS IN NONPARAMETRIC MONOTONE REGRESSION

• 成果类型: Article
• 署名作者: Chakraborty, Moumita; Ghosal, Subhashis
• 署名单位: North Carolina State University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/20-AOS1989
• 发表日期: 2021
• 页码: 1011-1028
• 关键词: von mises phenomenon; confidence bands; testing monotonicity; inference; density; THEOREMS

摘要

For nonparametric univariate regression under a monotonicity constraint on the regression function f, we study the coverage of a Bayesian credible interval for f (x(0)), where x(0) is an interior point. Analysis of the posterior becomes a lot more tractable by considering a projection-posterior distribution based on a finite random series of step functions with normal basis coefficients as a prior for f. A sample f from the resulting conjugate posterior distribution is projected on the space of monotone increasing functions to obtain a monotone function f* closest to f, inducing the projection-posterior. We use projection-posterior samples to obtain credible intervals for f (x(0)). We obtain the asymptotic coverage of the credible interval thus constructed and observe that it is free of nuisance parameters involving the true function. We observe a very interesting phenomenon that the coverage is typically higher than the nominal credibility level, the opposite of a phenomenon observed by Cox (Ann. Statist. 21 (1993) 903-923) in the Gaussian sequence model. We further show that a recalibration gives the right asymptotic coverage by starting from a lower credibility level that can be explicitly calculated.