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

TL;DR

This paper extends Kiefer-Wolfowitz type results to general monotone function estimation, providing convergence rates for Grenander-type estimators and demonstrating their asymptotic equivalence to kernel estimators.

Contribution

It generalizes the convergence rate analysis of Grenander estimators to a broad setting, including various monotone function estimation problems.

Findings

Supremum distance between estimator and naive estimator is of order $O_p(n^{-1} ext{log} n)^{2/(4- au)}$

In typical Gaussian cases, the convergence rate is $n^{-2/3}( ext{log} n)^{2/3}$

For the primitive of the estimator, the rate improves to $n^{-1} ext{log} n$

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 $\hat{F}_n$ of a naive estimator $F_n$ of the integrated curve $F$ corresponding to $f$. We prove that the supremum distance between $\hat{F}_n$ and $F_n$ is of the order $O_p(n^{-1}\log n)^{2/(4-\tau)}$, 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 is $n^{-2/3}(\log n)^{2/3}$ is in the same spirit as the one obtained by Kiefer and Wolfowitz (1976) 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 (2007). 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.