Nonparametric estimation of multivariate scale mixtures of uniform densities

TL;DR

This paper investigates the properties and consistency of maximum likelihood estimators for the class of multivariate scale mixtures of uniform densities, providing theoretical foundations and discussing convergence rates.

Contribution

It establishes the existence, uniqueness, and strong consistency of the MLE for the scale mixture of uniforms family, and derives a minimax lower bound for density estimation.

Findings

Proved existence of the MLE in the family

Established Fenchel characterizations of the MLE

Derived a minimax lower bound for density estimation

Abstract

Suppose that $\m{U} = (U_1, \ldots , U_d) $ has a Uniform$([0,1]^d)$ distribution, that $\m{Y} = (Y_1 , \ldots , Y_d) $ has the distribution $G$ on $\RR_+^d$, and let $\m{X} = (X_1 , \ldots , X_d) = (U_1 Y_1 , \ldots , U_d Y_d )$. The resulting class of distributions of $\m{X}$ (as $G$ varies over all distributions on $\RR_+^d$) is called the {\sl Scale Mixture of Uniforms} class of distributions, and the corresponding class of densities on $\RR_+^d$ is denoted by $\{\cal F}_{SMU}(d)$. We study maximum likelihood estimation in the family ${\cal F}_{SMU}(d)$. We prove existence of the MLE, establish Fenchel characterizations, and prove strong consistency of the almost surely unique maximum likelihood estimator (MLE) in ${\cal F}_{SMU}(d)$. We also provide an asymptotic minimax lower bound for estimating the functional $f \mapsto f(\m{x})$ under reasonable differentiability assumptions on $f\in{\cal F}_{SMU} (d)$ in a neighborhood of $\m{x}$. We conclude the paper with discussion, conjectures and open problems pertaining to global and local rates of convergence of the MLE.