Adaptivity and optimality of the monotone least-squares estimator

Abstract

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 locally asymptotic minimax, in the sense that, for each $f_0$, the attained rate of the probabilistic error is uniform over a shrinking $L^2$-neighborhood of $f_0$ and there is no estimator that attains a significantly better uniform rate over these shrinking neighborhoods. Therefore, it adapts to the individual underlying function, not to a smoothness class of functions. We also give general conditions for which we can calculate a (non-standard) limiting distribution for the estimator.