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 R + d , 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 R + d ) is called the Scale Mixture of Uniforms class of distributions, and the corresponding class of densities on R + d is denoted by F SMU ( 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 ↦ f ( x ) under reasonable differentiability assumptions on f ∈ 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.