  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.
