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dc.contributor.authorHan, Qiyang
dc.contributor.authorWang, Tengyao
dc.contributor.authorChatterjee, Sabyasachi
dc.contributor.authorSamworth, Richard
dc.date.accessioned2019-01-11T00:31:47Z
dc.date.available2019-01-11T00:31:47Z
dc.date.issued2019-08-03
dc.identifier.issn0090-5364
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/287832
dc.description.abstractWe 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.sponsorshipThe 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.publisherInstitute of Mathematical Statistics
dc.rightsAll rights reserved
dc.titleIsotonic regression in general dimensions
dc.typeArticle
prism.endingPage2471
prism.issueIdentifier5
prism.publicationDate2019
prism.publicationNameAnnals of Statistics
prism.startingPage2440
prism.volume47
dc.identifier.doi10.17863/CAM.35147
dcterms.dateAccepted2018-07-21
rioxxterms.versionofrecord10.1214/18-AOS1753
rioxxterms.versionVoR
rioxxterms.licenseref.urihttp://www.rioxx.net/licenses/all-rights-reserved
rioxxterms.licenseref.startdate2019-08-03
dc.contributor.orcidSamworth, Richard [0000-0003-2426-4679]
rioxxterms.typeJournal Article/Review
pubs.funder-project-idEngineering and Physical Sciences Research Council (EP/J017213/1)
pubs.funder-project-idLeverhulme Trust (PLP-2014-353)
pubs.funder-project-idEngineering and Physical Sciences Research Council (EP/N031938/1)
pubs.funder-project-idEngineering and Physical Sciences Research Council (EP/P031447/1)
cam.orpheus.successMon Jul 13 08:28:16 BST 2020 - The item has an open VoR version.
rioxxterms.freetoread.startdate2100-01-01


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