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 G on [Formula: see text], and let X = (X(1), … , X(d)) = (U(1)Y(1), … , U(d)Y(d)). The resulting class of distributions of X (as G varies over all distributions on [Formula: see text]) is called the Scale Mixture of Uniforms class of distributions, and the corresponding class of densities on [Formula: see text] is denoted by [Formula: see text]. We study maximum likelihood estimation in the family [Formula: see text]. We prove existence of the MLE, establish Fenchel characterizations, and prove strong consistency of the almost surely unique maximum likelihood estimator (MLE) in [Formula: see text]. We also provide an asymptotic minimax lower bound for estimating the functional f ↦ f(x) under reasonable differentiability assumptions on f ∈ [Formula: see text] 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.