Design, modelling, and analysis of some biological data sets
Design, data & analysis
Density estimation under qualitative assumptions in higher dimensions
Journal of Multivariate Analysis
Generalized MLE of a joint distribution function with multivariate interval-censored data
Journal of Multivariate Analysis
On the risk of estimates for block decreasing densities
Journal of Multivariate Analysis
Journal of Multivariate Analysis
Least squares estimation of a k-monotone density function
Computational Statistics & Data Analysis
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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"1Y"1,...,U"dY"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"S"M"U(d). We study maximum likelihood estimation in the family F"S"M"U(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"S"M"U(d). We also provide an asymptotic minimax lower bound for estimating the functional f@?f(x) under reasonable differentiability assumptions on f@?F"S"M"U(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.