Mathematics – Statistics Theory
Scientific paper
2010-05-08
Mathematics
Statistics Theory
39 pages, 4 figures
Scientific paper
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.
Pavlides Marios G.
Wellner Jon A.
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