Estimation of unimodal densities without smoothness assumptions

Abstract

The Grenander estimator of a decreasing density, which is defined as the derivative of the concave envelope of the empirical c.d.f., is known to be a very good estimator of an unknown decreasing density on the half-line $\mathbb{R}^+$ when this density is not assumed to be smooth. It is indeed the maximum likelihood estimator and one can get precise upper bounds for its risk when the loss is measured by the $\mathbb{L}^1$-distance between densities. Moreover, if one restricts oneself to the compact subsets of decreasing densities bounded by H with support on $[0, L]$ the risk of this estimator is within a fixed factor of the minimax risk. The same is true if one deals with the maximum likelihood estimator for unimodal densities with known mode. When the mode is unknown, the maximum likelihood estimator does not exist any more. We shall provide a general purpose estimator (together with a computational algorithm) for estimating nonsmooth unimodal densities. Its risk is the same as the risk of the Grenander estimator based on the knowledge of the true mode plus some lower order term. It can also cope with small departures from unimodality.