文章基本信息

标题：A Kiefer-Wolfowitz type of result in a general setting, with an application to smooth monotone estimation
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作者：Cécile Durot ; Hendrik P. Lopuhaä
期刊名称：Electronic Journal of Statistics
印刷版ISSN：1935-7524
出版年度：2014
卷号：8
期号：2
页码：2479-2513
DOI：10.1214/14-EJS958
语种：English
出版社：Institute of Mathematical Statistics
摘要：We consider Grenander type estimators for monotone functions $f$ in a very general setting, which includes estimation of monotone regression curves, monotone densities, and monotone failure rates. These estimators are defined as the left-hand slope of the least concave majorant $\widehat{F}_{n}$ of a naive estimator $F_{n}$ of the integrated curve $F$ corresponding to $f$. We prove that the supremum distance between $\widehat{F}_{n}$ and $F_{n}$ is of the order $O_{p}(n^{-1}\logn)^{2/(4-\tau)}$, for some $\tau\in[0,4)$ that characterizes the tail probabilities of an approximating process for $F_{n}$. In typical examples, the approximating process is Gaussian and $\tau=1$, in which case the convergence rate $n^{-2/3}(\log n)^{2/3}$ is in the same spirit as the one obtained by Kiefer and Wolfowitz [9] for the special case of estimating a decreasing density. We also obtain a similar result for the primitive of $F_{n}$, in which case $\tau=2$, leading to a faster rate $n^{-1}\log n$, also found by Wang and Woodfroofe [22]. As an application in our general setup, we show that a smoothed Grenander type estimator and its derivative are asymptotically equivalent to the ordinary kernel estimator and its derivative in first order.