Sharp oracle inequalities for Least Squares estimators in shape restricted regression

By Pierre C. Bellec. 2015

100 citations recorded by the source corpus.

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 $n^{-2/3}$ as well as a parametric rate of order $k/n$ up to logarithmic factors, where $k$ is the number of constant pieces of the true parameter. In univariate convex regression, the LS estimator satisfies an adaptive risk bound of order $q/n$ up to logarithmic factors, where $q$ 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 $n^{-4/5}$ for equispaced design points, we show that the nonparametric rate of convex regression can be as slow as $n^{-2/3}$ 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 $n^{-2/3}$.

Subfields: Computational Mechanics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/899ae6c8dcb81ab34bbc4d8aae8ac5ca8c7e81c2

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Citations in the corpus

Optimal rates of statistical seriation (2016, methodology): “Bellec in the revision of [10] that was prepared independently and contemporaneously to this paper.”
On matrix estimation under monotonicity constraints (2015, methodology): “Please see Bellec [3] for more background and general theory on such model misspecification oracle inequalities.”
Nonparametric Shape-Restricted Regression (2017, methodology): “Bellec [19] extended it to the case when θ∗ / ∈ C by proving Lemma 3.”