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August 2015 On risk bounds in isotonic and other shape restricted regression problems
Sabyasachi Chatterjee, Adityanand Guntuboyina, Bodhisattva Sen
Ann. Statist. 43(4): 1774-1800 (August 2015). DOI: 10.1214/15-AOS1324

Abstract

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.

Citation

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Sabyasachi Chatterjee. Adityanand Guntuboyina. Bodhisattva Sen. "On risk bounds in isotonic and other shape restricted regression problems." Ann. Statist. 43 (4) 1774 - 1800, August 2015. https://doi.org/10.1214/15-AOS1324

Information

Received: 1 May 2014; Revised: 1 February 2015; Published: August 2015
First available in Project Euclid: 17 June 2015

zbMATH: 1317.62032
MathSciNet: MR3357878
Digital Object Identifier: 10.1214/15-AOS1324

Subjects:
Primary: 62C20, 62G08

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 4 • August 2015
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