dc.contributor.author Balabdaoui, Fadoua dc.date.accessioned 2010-01-26T15:52:58Z dc.date.available 2010-01-26T15:52:58Z dc.date.issued 2007 dc.identifier.uri https://basepub.dauphine.fr/handle/123456789/3145 dc.language.iso en en dc.subject Convex density en dc.subject Hampel's problem en dc.subject estimation at the boundary en dc.subject Brownian motion en dc.subject.ddc 519 en dc.title Consistent estimation of a convex density at the origin en dc.type Article accepté pour publication ou publié dc.description.abstracten Motivated by Hampel's birds migration problem, \mycite{gjw:01b} established the asymptotic distribution theory for the nonparametric Least Squares and Maximum Likelihood estimators of a convex and decreasing density, $g_0$, at a fixed point $t_0 > 0$. However, estimation of the distribution function of the birds' resting times involves estimation of $g'_0$ at 0, a boundary point at which the estimators are not consistent. In this paper, we focus on the Least Squares estimator, $\tilde{g}_n$. Our goal is to show that consistent estimators of both $g_0(0)$ and $g'_0(0)$ can be based solely on $\tilde{g}_n$. Following the idea of \mycite{kuliandlopuh:06} in monotone estimation, we show that it suffices to take $\tilde{g}_n(n^{-\alpha})$ and $\tilde{g}'_n(n^{-\alpha})$, with $\alpha \in (0,1/3)$. We establish their joint asymptotic distributions and show that $\alpha =1/5$ should be taken as it yields the fastest rates of convergence. en dc.relation.isversionofjnlname Mathematical Methods of Statistics dc.relation.isversionofjnlvol 16 en dc.relation.isversionofjnlissue 2 en dc.relation.isversionofjnldate 2007 dc.relation.isversionofjnlpages 77-95 en dc.relation.isversionofdoi http://dx.doi.org/10.3103/S1066530707020019 en dc.description.sponsorshipprivate oui en dc.relation.isversionofjnlpublisher Allerton Press, Inc. en dc.subject.ddclabel Probabilités et mathématiques appliquées en
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