Translator Disclaimer
October 2019 Isotonic regression in general dimensions
Qiyang Han, Tengyao Wang, Sabyasachi Chatterjee, Richard J. Samworth
Ann. Statist. 47(5): 2440-2471 (October 2019). DOI: 10.1214/18-AOS1753

Abstract

We study the least squares regression function estimator over the class of real-valued functions on $[0,1]^{d}$ that are increasing in each coordinate. For uniformly bounded signals and with a fixed, cubic lattice design, we establish that the estimator achieves the minimax rate of order $n^{-\min\{2/(d+2),1/d\}}$ in the empirical $L_{2}$ loss, up to polylogarithmic factors. Further, we prove a sharp oracle inequality, which reveals in particular that when the true regression function is piecewise constant on $k$ hyperrectangles, the least squares estimator enjoys a faster, adaptive rate of convergence of $(k/n)^{\min(1,2/d)}$, again up to polylogarithmic factors. Previous results are confined to the case $d\leq2$. Finally, we establish corresponding bounds (which are new even in the case $d=2$) in the more challenging random design setting. There are two surprising features of these results: first, they demonstrate that it is possible for a global empirical risk minimisation procedure to be rate optimal up to polylogarithmic factors even when the corresponding entropy integral for the function class diverges rapidly; second, they indicate that the adaptation rate for shape-constrained estimators can be strictly worse than the parametric rate.

Citation

Download Citation

Qiyang Han. Tengyao Wang. Sabyasachi Chatterjee. Richard J. Samworth. "Isotonic regression in general dimensions." Ann. Statist. 47 (5) 2440 - 2471, October 2019. https://doi.org/10.1214/18-AOS1753

Information

Received: 1 August 2017; Revised: 1 April 2018; Published: October 2019
First available in Project Euclid: 3 August 2019

zbMATH: 07114918
MathSciNet: MR3988762
Digital Object Identifier: 10.1214/18-AOS1753

Subjects:
Primary: 62C20, 62G05, 62G08

Rights: Copyright © 2019 Institute of Mathematical Statistics

JOURNAL ARTICLE
32 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.47 • No. 5 • October 2019
Back to Top