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dc.contributor.authorBalabdaoui, Fadoua
dc.contributor.authorWellner, Jon
dc.date.accessioned2010-02-15T11:01:36Z
dc.date.available2010-02-15T11:01:36Z
dc.date.issued2007
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/3455
dc.descriptionhttp://projecteuclid.org/euclid.aos/1201012971en
dc.language.isoenen
dc.subjectasymptotic distributionen
dc.subjectLSEen
dc.subjectMLEen
dc.subjectcompletely monotoneen
dc.subjectk-monotoneen
dc.subjectsplinesen
dc.subject.ddc519en
dc.titleEstimation of a k-monotone density: limit distribution theory and the Spline connectionen
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenWe study the asymptotic behavior of the Maximum Likelihood and Least Squares estimators of a $k-$monotone density $g_0$ at a fixed point $x_0$ when $k > 2$. In \mycite{balabwell:04a}, it was proved that both estimators exist and are splines of degree $k-1$ with simple knots. These knots, which are also the jump points of the $(k-1)-$st derivative of the estimators, cluster around a point $x_0 > 0$ under the assumption that $g_0$ has a continuous $k$-th derivative in a neighborhood of $x_0$ and $(-1)^k g^{(k)}_0(x_0) > 0$. If $\tau^{-}_n$ and $\tau^{+}_n$ are two successive knots, we prove that the random ``gap'' \ $\tau^{+}_n - \tau^{-}_n $ is $O_p(n^{-1/(2k+1)})$ for any $k > 2$ if a conjecture about the upper bound on the error in a particular Hermite interpolation via odd-degree splines holds. Based on the order of the gap, the asymptotic distribution of the Maximum Likelihood and Least Squares estimators can be established. We find that the $j-$th derivative of the estimators at $x_0$ converges at the rate $n^{-(k-j)/(2k+1)}$ for $j=0, \ldots, k-1$. The limiting distribution depends on an almost surely uniquely defined stochastic process $H_k$ that stays above (below) the $k$-fold integral of Brownian motion plus a deterministic drift, when $k$ is even (odd).en
dc.relation.isversionofjnlnameAnnals of Statistics
dc.relation.isversionofjnlvol35en
dc.relation.isversionofjnlissue6en
dc.relation.isversionofjnldate2007
dc.relation.isversionofjnlpages2536-2564en
dc.relation.isversionofdoihttp://dx.doi.org/10.1214/009053607000000262en
dc.identifier.urlsitehttp://hal.archives-ouvertes.fr/hal-00363240/en/en
dc.description.sponsorshipprivateouien
dc.relation.isversionofjnlpublisherInstitute of Mathematical Statisticsen
dc.subject.ddclabelProbabilités et mathématiques appliquéesen


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