THE LIMIT DISTRIBUTION OF THE L∞-ERROR OF GRENANDER-TYPE ESTIMATORS

• 成果类型: Article
• 署名作者: Durot, Cecile; Kulikov, Vladimir N.; Lopuhaa, Hendrik P.
• 署名单位: Delft University of Technology
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/12-AOS1015
• 发表日期: 2012
• 页码: 1578-1608
• 关键词: Asymptotic Normality; monotone density; smooth monotone; brownian-motion

摘要

Let f be a nonincreasing function defined on [0, 1]. Under standard regularity conditions, we derive the asymptotic distribution of the supremum norm of the difference between f and its Grenander-type estimator on sub-intervals of [0, 1]. The rate of convergence is found to be of order (n/logn)(-1/3) and the limiting distribution to be Gumbel.