dc.contributor.author Han, Qiyang en dc.contributor.author Wang, Tengyao en dc.contributor.author Chatterjee, Sabyasachi en dc.contributor.author Samworth, Richard en dc.date.accessioned 2019-01-11T00:31:47Z dc.date.available 2019-01-11T00:31:47Z dc.date.issued 2019-08-03 en dc.identifier.issn 0090-5364 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/287832 dc.description.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 poly-logarithmic 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 poly-logarithmic factors. Previous results are confined to the case $d \leq 2$. 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 poly-logarithmic 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. dc.description.sponsorship The research of the first author is supported in part by NSF Grant DMS-1566514. The research of the second and fourth authors is supported by EPSRC fellowship EP/J017213/1 and a grant from the Leverhulme Trust RG81761. dc.publisher Institute of Mathematical Statistics dc.rights All rights reserved dc.title Isotonic regression in general dimensions en dc.type Article prism.endingPage 2471 prism.issueIdentifier 5 en prism.publicationDate 2019 en prism.publicationName Annals of Statistics en prism.startingPage 2440 prism.volume 47 en dc.identifier.doi 10.17863/CAM.35147 dcterms.dateAccepted 2018-07-21 en rioxxterms.version VoR rioxxterms.licenseref.uri http://www.rioxx.net/licenses/all-rights-reserved en rioxxterms.licenseref.startdate 2019-08-03 en dc.contributor.orcid Samworth, Richard [0000-0003-2426-4679] rioxxterms.type Journal Article/Review en pubs.funder-project-id EPSRC (EP/J017213/1) pubs.funder-project-id Leverhulme Trust (PLP-2014-353) pubs.funder-project-id LANCASTER UNIVERSITY (FB EPSRC) (EP/N031938/1) pubs.funder-project-id EPSRC (EP/P031447/1) cam.orpheus.success Mon Jul 13 08:28:16 BST 2020 - The item has an open VoR version. * rioxxterms.freetoread.startdate 2100-01-01
﻿