RELAXING THE GAUSSIAN ASSUMPTION IN SHRINKAGE AND SURE IN HIGH DIMENSION

• 成果类型: Article
• 署名作者: Fathi, Max; Goldstein, Larry; Reinert, Gesine; Saumard, Adrien
• 署名单位: Universite Paris Cite; Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite; Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite; University of Southern California; University of Oxford; Ecole Nationale de la Statistique et de l'Analyse de l'Information (ENSAI); Universite de Rennes
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/22-AOS2208
• 发表日期: 2022
• 页码: 2737-2766
• 关键词: stein kernels; bounds; estimators; noise

摘要

Shrinkage estimation is a fundamental tool of modern statistics, pioneered by Charles Stein upon his discovery of the famous paradox involving the multivariate Gaussian. A large portion of the subsequent literature only considers the efficiency of shrinkage, and that of an associated procedure known as Stein's Unbiased Risk Estimate, or SURE, in the Gaussian setting of that original work. We investigate what extensions to the domain of validity of shrinkage and SURE can be made away from the Gaussian through the use of tools developed in the probabilistic area now known as Stein's method. We show that shrinkage is efficient away from the Gaussian under very mild conditions on the distribution of the noise. SURE is also proved to be adaptive under similar assumptions, and in particular in a way that retains the classical asymptotics of Pinsker's theorem. Notably, shrinkage and SURE are shown to be efficient under mild distributional assumptions, and particularly for general isotropic log-concave measures.