Maximum likelihood estimation of smooth monotone and unimodal densities

摘要

We study the nonparametric estimation of univariate monotone and unimodal densities using the maximum smoothed likelihood approach. The monotone estimator is the derivative of the least concave majorant of the distribution corresponding to a kernel estimator. We prove that the mapping on distributions Phi with density phi, phi bar right arrow the derivative of the least concave majorant of Phi, is a contraction in all L-P norms (1 less than or equal to p less than or equal to infinity), 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. [References: 35]