Adaptation in Some Shape-Constrained Regression Problems

Adityanand Guntuboyina

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.

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