Open Access
February 2019 Adaptive risk bounds in unimodal regression
Sabyasachi Chatterjee, John Lafferty
Bernoulli 25(1): 1-25 (February 2019). DOI: 10.3150/16-BEJ922


We study the statistical properties of the least squares estimator in unimodal sequence estimation. Although closely related to isotonic regression, unimodal regression has not been as extensively studied. We show that the unimodal least squares estimator is adaptive in the sense that the risk scales as a function of the number of values in the true underlying sequence. Such adaptivity properties have been shown for isotonic regression by Chatterjee et al. (Ann. Statist. 43 (2015) 1774–1800) and Bellec (Sharp oracle inequalities for Least Squares estimators in shape restricted regression (2016)). A technical complication in unimodal regression is the non-convexity of the underlying parameter space. We develop a general variational representation of the risk that holds whenever the parameter space can be expressed as a finite union of convex sets, using techniques that may be of interest in other settings.


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Sabyasachi Chatterjee. John Lafferty. "Adaptive risk bounds in unimodal regression." Bernoulli 25 (1) 1 - 25, February 2019.


Received: 1 July 2016; Revised: 1 December 2016; Published: February 2019
First available in Project Euclid: 12 December 2018

zbMATH: 07007197
MathSciNet: MR3892309
Digital Object Identifier: 10.3150/16-BEJ922

Keywords: isotonic regression , minimax bounds , shape constrained inference , unimodal regression

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 1 • February 2019
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