VARIATIONAL ANALYSIS OF CONSTRAINED M-ESTIMATORS

• 成果类型: Article
• 署名作者: Royset, Johannes O.; Wets, Roger J-B
• 署名单位: United States Department of Defense; United States Navy; Naval Postgraduate School; University of California System; University of California Davis
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/19-AOS1905
• 发表日期: 2020
• 页码: 2759-2790
• 关键词: maximum-likelihood-estimation; monotone approximation; CONVERGENCE; Consistency; optimization

摘要

We propose a unified framework for establishing existence of nonparametric M-estimators, computing the corresponding estimates, and proving their strong consistency when the class of functions is exceptionally rich. In particular, the framework addresses situations where the class of functions is complex involving information and assumptions about shape, pointwise bounds, location of modes, height at modes, location of level-sets, values of moments, size of subgradients, continuity, distance to a prior function, multivariate total positivity and any combination of the above. The class might be engineered to perform well in a specific setting even in the presence of little data. The framework views the class of functions as a subset of a particular metric space of upper semicontinuous functions under the Attouch-Wets distance. In addition to allowing a systematic treatment of numerous M-estimators, the framework yields consistency of plug-in estimators of modes of densities, maximizers of regression functions, level-sets of classifiers and related quantities, and also enables computation by means of approximating parametric classes. We establish consistency through a one-sided law of large numbers, here extended to sieves, that relaxes assumptions of uniform laws, while ensuring global approximations even under model misspecification.