ON ADAPTIVE INFERENCE AND CONFIDENCE BANDS

• 成果类型: Article
• 署名作者: Hoffmann, Marc; Nickl, Richard
• 署名单位: Institut Polytechnique de Paris; ENSAE Paris; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris-Est-Creteil-Val-de-Marne (UPEC); University of Cambridge
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/11-AOS903
• 发表日期: 2011
• 页码: 2383-2409
• 关键词: sup-norm loss; DENSITY-ESTIMATION; posterior distributions; wavelet; estimators; regression; rates; intervals; balls

摘要

The problem of existence of adaptive confidence bands for an unknown density f that belongs to a nested scale of Holder classes over R or [0, 1] is considered. Whereas honest adaptive inference in this problem is impossible already for a pair of Holder balls Sigma (r), Sigma(s), r not equal s, of fixed radius, a non-parametric distinguishability condition is introduced under which adaptive confidence bands can be shown to exist. It is further shown that this condition is necessary and sufficient for the existence of honest asymptotic confidence bands, and that it is strictly weaker than similar analytic conditions recently employed in Gine and Nickl [Ann. Statist. 38 (2010) 1122-1170]. The exceptional sets for which honest inference is not possible have vanishingly small probability under natural priors on Holder balls Sigma (s). If no upper bound for the radius of the Holder balls is known, a price for adaptation has to be paid, and near-optimal adaptation is possible for standard procedures. The implications of these findings for a general theory of adaptive inference are discussed.