  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.
