LIMIT DISTRIBUTION THEORY FOR MAXIMUM LIKELIHOOD ESTIMATION OF A LOG-CONCAVE DENSITY

• 成果类型: Article
• 署名作者: Balabdaoui, Fadoua; Rufibach, Kaspar; Wellner, Jon A.
• 署名单位: Universite PSL; Universite Paris-Dauphine; University of Zurich; University of Washington; University of Washington Seattle; University of Gottingen
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/08-AOS609
• 发表日期: 2009
• 页码: 1299-1331
• 关键词: symmetric unimodal density; asymptotic distributions; smoothness assumptions; kernel estimators; Mode Estimation; restrictions; probability

摘要

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form f(0) = exp phi(0) where phi(0) is a concave function on R. The pointwise limiting distributions depend on the second and third derivatives at 0 of H-k, the lower invelope of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of phi(0) = log f(0) at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode M(f(0)) and establish a new local asymptotic minimax lower bound which shows the optimality Of Our mode estimator in terms of both rate of convergence and dependence of constants on Population values.