Maximum likelihood estimation of smooth monotone and unimodal densities

Abstract

We study the nonparametric estimation of univariate monotone and unimodal densities usingthe maximum smoothed likelihood approach. The monotone estimator is the derivative of the least concave majorant of the distribution correspondingto a kernel estimator.We prove that the mapping on distributions $\Phi$ with density $\varphi$, $$\varphi \mapsto \text{the derivative of the least concave majorant of $\Phi},$$ is a contraction in all $L^P$ norms $(1 \leq p \leq \infty)$, and some other “distances” such as the Hellinger and Kullback–Leibler distances. The contractivity implies error bounds for monotone density estimation. Almost the same error bounds hold for unimodal estimation.