The Annals of Statistics

On risk bounds in isotonic and other shape restricted regression problems

Sabyasachi Chatterjee, Adityanand Guntuboyina, and Bodhisattva Sen

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We consider the problem of estimating an unknown $\theta\in\mathbb{R}^{n}$ from noisy observations under the constraint that $\theta$ belongs to certain convex polyhedral cones in $\mathbb{R}^{n}$. Under this setting, we prove bounds for the risk of the least squares estimator (LSE). The obtained risk bound behaves differently depending on the true sequence $\theta$ which highlights the adaptive behavior of $\theta$. As special cases of our general result, we derive risk bounds for the LSE in univariate isotonic and convex regression. We study the risk bound in isotonic regression in greater detail: we show that the isotonic LSE converges at a whole range of rates from $\log n/n$ (when $\theta$ is constant) to $n^{-2/3}$ (when $\theta$ is uniformly increasing in a certain sense). We argue that the bound presents a benchmark for the risk of any estimator in isotonic regression by proving nonasymptotic local minimax lower bounds. We prove an analogue of our bound for model misspecification where the true $\theta$ is not necessarily nondecreasing.

Article information

Ann. Statist., Volume 43, Number 4 (2015), 1774-1800.

Received: May 2014
Revised: February 2015
First available in Project Euclid: 17 June 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression 62C20: Minimax procedures

Adaptation convex polyhedral cones global risk bounds local minimax bounds model misspecification statistical dimension


Chatterjee, Sabyasachi; Guntuboyina, Adityanand; Sen, Bodhisattva. On risk bounds in isotonic and other shape restricted regression problems. Ann. Statist. 43 (2015), no. 4, 1774--1800. doi:10.1214/15-AOS1324.

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Supplemental materials

  • Supplement to “On risk bounds in isotonic and other shape restricted regression problems”. In the supplementary paper [13] we provide the proofs of Lemmas 2.4, 2.5, 2.6, 6.3 and 6.4 and Theorems 4.1, 4.2, 4.3, 5.3 and 6.1. We also state and prove Lemma 11.1, which is used in the proof of Theorem 4.1, and Lemma 11.2, which is used in the proof of Theorem 3.1.