Adaptive confidence sets in L2

• 成果类型: Article
• 署名作者: Bull, Adam D.; Nickl, Richard
• 署名单位: University of Cambridge
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-012-0446-z
• 发表日期: 2013
• 页码: 889-919
• 关键词: Concentration Inequalities; density; wavelet; rates

摘要

The problem of constructing confidence sets that are adaptive in -loss over a continuous scale of Sobolev classes of probability densities is considered. Adaptation holds, where possible, with respect to both the radius of the Sobolev ball and its smoothness degree, and over maximal parameter spaces for which adaptation is possible. Two key regimes of parameter constellations are identified: one where full adaptation is possible, and one where adaptation requires critical regions be removed. Techniques used to derive these results include a general nonparametric minimax test for infinite-dimensional null- and alternative hypotheses, and new lower bounds for -adaptive confidence sets.