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

Abstract

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.