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Sharp oracle inequalities for Least Squares estimators in shape restricted regression

Abstract

The performance of Least Squares (LS) estimators is studied in isotonic, unimodal and convex regression. Our results have the form of sharp oracle inequalities that account for the model misspecification error. In isotonic and unimodal regression, the LS estimator achieves the nonparametric rate n2/3n^{-2/3} as well as a parametric rate of order k/nk/n up to logarithmic factors, where kk is the number of constant pieces of the true parameter. In univariate convex regression, the LS estimator satisfies an adaptive risk bound of order q/nq/n up to logarithmic factors, where qq is the number of affine pieces of the true regression function. This adaptive risk bound holds for any design points. While Guntuboyina and Sen (2013) established that the nonparametric rate of convex regression is of order n4/5n^{-4/5} for equispaced design points, we show that the nonparametric rate of convex regression can be as slow as n2/3n^{-2/3} for some worst-case design points. This phenomenon can be explained as follows: Although convexity brings more structure than unimodality, for some worst-case design points this extra structure is uninformative and the nonparametric rates of unimodal regression and convex regression are both n2/3n^{-2/3}.

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