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We consider Grenander type estimators for monotone functions $f$ in a verygeneral setting, which includes estimation of monotone regression curves,monotone densities, and monotone failure rates. These estimators are defined asthe left-hand slope of the least concave majorant $\hat{F}_n$ of a naiveestimator $F_n$ of the integrated curve $F$ corresponding to $f$. We prove thatthe 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 characterizesthe tail probabilities of an approximating process for $F_n$. In typicalexamples, the approximating process is Gaussian and $\tau=1$, in which case theconvergence rate is $n^{-2/3}(\log n)^{2/3}$ is in the same spirit as the oneobtained by Kiefer and Wolfowitz (1976) for the special case of estimating adecreasing 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 byWang and Woodfroofe (2007). As an application in our general setup, we showthat a smoothed Grenander type estimator and its derivative are asymptoticallyequivalent to the ordinary kernel estimator and its derivative in first order.

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