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

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 $O_p(n^{-2/3}(\log n)^{2/3})$. This result is in the same spirit as the one obtained by Kiefer and Wolfowitz (1976) for the special case of estimating a decreasing density. As an application in our general setup, we show that a smoothed Grenander type estimator is asymptotically equivalent to the ordinary kernel estimator in first order.