Adaptation in some shapeconstrained regression problems
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We consider the problem of estimating a normal mean constrained to be in a convex
polyhedral cone in Euclidean space. We say that the true mean is sparse if it
belongs to a low dimensional face of the cone. We show that, in a certain natural
subclass of these problems, the maximum likelihood estimator automatically adapts
to sparsity in the underlying true mean. We discuss the problems of convex
regression and univariate and bivariate isotonic regression as examples.
This talk is part of the Probability Theory and Statistics in High and Infinite Dimensions series.
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