  In this paper we will consider the estimation of a monotone regression (or
density) function in a fixed point by the least squares (Grenander) estimator.
We will show that this estimator is fully adaptive, in the sense that the
attained rate is given by a functional relation using the underlying function
$f_0$, and not by some smoothness parameter, and that this rate is optimal when
considering the class of all monotone functions, in the sense that there exists
a sequence of alternative monotone functions $f_1$, such that no other
estimator can attain a better rate for both $f_0$ and $f_1$. We also show that
under mild conditions the estimator attains the same rate in $L^q$ sense, and
we give general conditions for which we can calculate a (non-standard) limiting
distribution for the estimator.
