Adaptive risk bounds in unimodal regression

Sabyasachi Chatterjee, John Lafferty

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)1-25
Number of pages25
JournalBernoulli
Volume25
Issue number1
DOIs
StatePublished - Feb 1 2019

Fingerprint

Least Squares Estimator
Regression
Isotonic Regression
Parameter Space
Oracle Inequalities
Non-convexity
Sharp Inequality
Adaptivity
Complications
Convex Sets
Statistical property
Union

Keywords

  • Isotonic regression
  • Minimax bounds
  • Shape constrained inference
  • Unimodal regression

ASJC Scopus subject areas

  • Statistics and Probability

Cite this

Adaptive risk bounds in unimodal regression. / Chatterjee, Sabyasachi; Lafferty, John.

In: Bernoulli, Vol. 25, No. 1, 01.02.2019, p. 1-25.

Research output: Contribution to journalArticle

Chatterjee, Sabyasachi ; Lafferty, John. / Adaptive risk bounds in unimodal regression. In: Bernoulli. 2019 ; Vol. 25, No. 1. pp. 1-25.
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