Maximum Likelihood Estimation of a Multi-Dimensional Log-Concave Density

Abstract

SummaryLet X1,…,Xn be independent and identically distributed random vectors with a (Lebesgue) density f. We first prove that, with probability 1, there is a unique log-concave maximum likelihood estimator f^n of f. The use of this estimator is attractive because, unlike kernel density estimation, the method is fully automatic, with no smoothing parameters to choose. Although the existence proof is non-constructive, we can reformulate the issue of computing f^n in terms of a non-differentiable convex optimization problem, and thus combine techniques of computational geometry with Shor’s r-algorithm to produce a sequence that converges to f^n. An R version of the algorithm is available in the..