Nonparametric estimation of multivariate scale mixtures of uniform densities

Suppose that U=(U1,…,Ud) has a Uniform([0,1]d) distribution, that Y=(Y1,…,Yd) has the distribution G on R+d, and let X=(X1,…,Xd)=(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 FSMU(d). We study maximum likelihood estimation in the family FSMU(d). We prove existence of the MLE, establish Fenchel characterizations, and prove strong consistency of the almost surely unique maximum likelihood estimator (MLE) in FSMU(d). We also provide an asymptotic minimax lower bound for estimating the functional f↦f(x) under reasonable differentiability assumptions on f∈FSMU(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.