A new method for estimation and model selection: ρ-estimation

• 成果类型: Article
• 署名作者: Baraud, Y.; Birge, L.; Sart, M.
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Cote d'Azur; Centre National de la Recherche Scientifique (CNRS); Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0673-5
• 发表日期: 2017
• 页码: 425-517
• 关键词: maximum-likelihood; Robust Estimation; risk bounds; rates; CONVERGENCE; inequalities

摘要

The aim of this paper is to present a new estimation procedure that can be applied in various statistical frameworks including density and regression and which leads to both robust and optimal (or nearly optimal) estimators. In density estimation, they asymptotically coincide with the celebrated maximum likelihood estimators at least when the statistical model is regular enough and contains the true density to estimate. For very general models of densities, including non-compact ones, these estimators are robust with respect to the Hellinger distance and converge at optimal rate (up to a possible logarithmic factor) in all cases we know. In the regression setting, our approach improves upon the classical least squares in many respects. In simple linear regression for example, it provides an estimation of the coefficients that are both robust to outliers and simultaneously rate-optimal (or nearly rate-optimal) for a large class of error distributions including Gaussian, Laplace, Cauchy and uniform among others.