Nonparametric estimation of multivariate scale mixtures of uniform densities

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