dc.contributor.author Balabdaoui, Fadoua dc.contributor.author Rufibach, Kaspar dc.contributor.author Wellner, Jon dc.date.accessioned 2010-02-15T10:43:26Z dc.date.available 2010-02-15T10:43:26Z dc.date.issued 2009 dc.identifier.uri https://basepub.dauphine.fr/handle/123456789/3454 dc.description http://projecteuclid.org/euclid.aos/1239369023 en dc.language.iso en en dc.subject asymptotic distribution en dc.subject log--concave density estimation en dc.subject integral of Brownian motion en dc.subject maximum likelihood en dc.subject nonparametric estimation en dc.subject shape constraints en dc.subject.ddc 519 en dc.title Limit distribution theory for maximum likelihood estimation of a log-concave density en dc.type Article accepté pour publication ou publié dc.description.abstracten We study the asymptotic behavior of the Maximum Likelihood estimator (MLE) of a density $f_0 = \exp \varphi_0$ where $\varphi_0$ is a concave function on $\mathbb{R}$. Existence, form, characterizations and uniform rate of convergence of this so-called log--concave density estimator are given in \cite{rufibach_06_diss} and \cite{duembgen_rufibach_06}. It turns out that the problem of identifying the limiting distribution of the estimator is connected to that of the least squares estimator of a convex density on $[0, \infty)$, since in both estimation problems a specific characterization of the estimators in terms of distribution functions is (up to sign) the same. We find that the limiting local behavior depends on the \corr{second and third derivatives} at $0$ of $H_k$, the outer envelope of the integrated Brownian motion process plus a drift term depending on the number of vanishing derivatives of the true log--concave density at the estimation point. Furthermore, we establish the limiting distribution for the weak convergence of the first location of the maximum of the MLE to the true mode. Numerical simulations using the R--package \cite{logcondens} were performed to generate samples from the limiting distributions and calculate estimates of their extreme quantiles. en dc.relation.isversionofjnlname Annals of Statistics dc.relation.isversionofjnlvol 37 en dc.relation.isversionofjnlissue 3 en dc.relation.isversionofjnldate 2009 dc.relation.isversionofjnlpages 1299-1331 en dc.relation.isversionofdoi http://dx.doi.org/10.1214/08-AOS609 en dc.identifier.urlsite http://hal.archives-ouvertes.fr/hal-00363228/en/ en dc.description.sponsorshipprivate oui en dc.relation.isversionofjnlpublisher Institute of Mathematical Statistics en dc.subject.ddclabel Probabilités et mathématiques appliquées en
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