Electronic Journal of Statistics

A Kiefer-Wolfowitz type of result in a general setting, with an application to smooth monotone estimation

Cécile Durot and Hendrik P. Lopuhaä

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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.

Article information

Electron. J. Statist., Volume 8, Number 2 (2014), 2479-2513.

First available in Project Euclid: 9 December 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62G07: Density estimation 62G08: Nonparametric regression 62N02: Estimation

Isotonic regression least concave majorant monotone density monotone failure rate monotone regression


Durot, Cécile; Lopuhaä, Hendrik P. A Kiefer-Wolfowitz type of result in a general setting, with an application to smooth monotone estimation. Electron. J. Statist. 8 (2014), no. 2, 2479--2513. doi:10.1214/14-EJS958. https://projecteuclid.org/euclid.ejs/1418134261

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