Show simple item record

dc.contributor.authorKim, Arlene KHen
dc.contributor.authorGuntuboyina, Adityananden
dc.contributor.authorSamworth, Richarden
dc.date.accessioned2017-09-28T16:48:07Z
dc.date.available2017-09-28T16:48:07Z
dc.date.issued2018-10en
dc.identifier.issn0090-5364
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/267439
dc.description.abstractThe log-concave maximum likelihood estimator of a density on the real line based on a sample of size n is known to attain the minimax optimal rate of convergence of O(n −4/5 ) with respect to, e.g., squared Hellinger distance. In this paper, we show that it also enjoys attractive adaptation properties, in the sense that it achieves a faster rate of convergence when the logarithm of the true density is k-affine (i.e. made up of k affine pieces), or close to k-affine, provided in each case that k is not too large. Our results use two different techniques: the first relies on a new Marshall’s inequality for log-concave density estimation, and reveals that when the true density is close to log-linear on its support, the log-concave maximum likelihood estimator can achieve the parametric rate of convergence in total variation distance. Our second approach depends on local bracketing entropy methods, and allows us to prove a sharp oracle inequality, which implies in particular a risk bound with respect to various global loss functions, including Kullback–Leibler divergence, of O k n log5/4 (en/k) when the true density is log-concave and its logarithm is close to k-affine.
dc.description.sponsorshipAKH Kim: National Research Foundation of Korea (NRF) grant 2017R1C1B5017344. A Guntuboyina: NSF Grant DMS-1309356. RJ Samworth: EPSRC Early Career Fellowship and a grant from the Leverhulme Trust.
dc.publisherInstitute of Mathematical Statistics
dc.subjectAdaptationen
dc.subjectbracketing entropyen
dc.subjectlog-concavityen
dc.subjectmaximum likelihood estimationen
dc.subjectMarshall's inequalityen
dc.titleADAPTATION IN LOG-CONCAVE DENSITY ESTIMATIONen
dc.typeArticle
prism.endingPage2306
prism.issueIdentifier5en
prism.publicationDate2018en
prism.publicationNameANNALS OF STATISTICSen
prism.startingPage2279
prism.volume46en
dc.identifier.doi10.17863/CAM.11980
dcterms.dateAccepted2017-07-24en
rioxxterms.versionofrecord10.1214/17-AOS1619en
rioxxterms.versionAM*
rioxxterms.licenseref.urihttp://www.rioxx.net/licenses/all-rights-reserveden
rioxxterms.licenseref.startdate2018-10en
dc.contributor.orcidSamworth, Richard [0000-0003-2426-4679]
rioxxterms.typeJournal Article/Reviewen
pubs.funder-project-idEPSRC (EP/J017213/1)
pubs.funder-project-idLeverhulme Trust (PLP-2014-353)
pubs.funder-project-idLANCASTER UNIVERSITY (FB EPSRC) (EP/N031938/1)
pubs.funder-project-idAlan Turing Institute (unknown)
cam.orpheus.successThu Jan 30 13:04:48 GMT 2020 - The item has an open VoR version.*
rioxxterms.freetoread.startdate2100-01-01


Files in this item

Thumbnail
Thumbnail
Thumbnail

This item appears in the following Collection(s)

Show simple item record