Nonparametric estimation of multivariate scale mixtures of uniform densities

Suppose that U = (U-1 ,..., U-d) has a Uniform([0. 1](d)) distribution, that Y = (Y-1 ,..., Y-d) has the distribution Con R-+(d), and let X = (X-1 ,..., X-d) (U1Y1 ,..., UdYd). The resulting class of distributions of X (as G varies over all distributions on R-+(d)) is called the Scale Mixture of Uniforms class of distributions, and the corresponding class of densities on R-+(d) is denoted by,MU (d). We study maximum likelihood estimation in the family F-SMU(d). We prove existence of the MLE, establish Fenchel characterizations, and prove strong consistency of the almost surely unique maximum likelihood estimator (MLE) in F-SMU (d). We also provide an asymptotic minimax lower bound for estimating the functional f bar right arrow (x) under reasonable differentiability assumptions on f is an element of F-SMU (d) in a neighborhood of x. We conclude the paper with discussion, conjectures and open problems pertaining to global and local rates of convergence of the MLE. (C) 2012 Elsevier Inc. All rights reserved.