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In this paper we will consider the estimation of a monotone regression (ordensity) function in a fixed point by the least squares (Grenander) estimator.We will show that this estimator is fully adaptive, in the sense that theattained 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 whenconsidering the class of all monotone functions, in the sense that there existsa sequence of alternative monotone functions $f_1$, such that no otherestimator can attain a better rate for both $f_0$ and $f_1$. We also show thatunder mild conditions the estimator attains the same rate in $L^q$ sense, andwe give general conditions for which we can calculate a (non-standard) limitingdistribution for the estimator.

Technical report: Adaptivity and optimality of the monotone least squares estimator for four different models