The Grenader Estimator: A Nonasymptotic Approach

Abstract

In this paper we shall investigate some nonasymptotic properties of the Grenander estimator of a decreasing density $f$. This estimator is defined as the slope of the smallest concave majorant of the empirical c.d.f. It will be proved that its risk, measured with $\mathbb{L}^1$-loss, is bounded by some functional depending on $f$ and the number $n$ of observations. For classes of uniformly bounded densities with a common compact support, upper bounds for the functional are shown to agree with older results about the minimax risk over these classes. The asymptotic behavior of the functional as $n$ goes to infinity is also in accordance with the known asymptotic performances of the Grenander estimator.