Almost sure convergence of the $ {k}_{T}$-occupation time density estimator

Boris Labrador

doi:10.1016/j.crma.2006.10.015

Statistics/Probability Theory

Almost sure convergence of the

k_{T}

-occupation time density estimator
[Convergence presque sûre de l'estimateur de la densité du

k_{T}

-temps d'occupation]

Présenté par : Paul Deheuvels

Boris Labrador ¹

¹ L.S.T.A., université Paris 6, 175, rue du Chevaleret, 75013 Paris, France

Comptes Rendus. Mathématique, Volume 343 (2006) no. 10, pp. 665-669.

Abstract (VO)
Résumé

In this Note, we introduce an extension of the k-nearest neighbor estimator in continuous time, the $k_{T}$ -occupation time estimator, and we give sufficient conditions for its existence. Then, we show the almost sure convergence for α-mixing and bounded processes in two cases, the superoptimal case (when parametric rates are reached) and the optimal case (when i.i.d. rates of density estimation are reached).

Dans cette Note, nous introduisons une extension de l'estimateur des k-plus proches voisins en temps continu, l'estimateur du $k_{T}$ -temps d'occupation, puis nous donnons des conditions d'existence de cet estimateur. Nous établissons également la convergence presque sûre pour des processus bornés α-mélangeants dans deux cas, le cas suroptimal (où la vitesse paramétrique est atteinte) et le cas optimal (où la vitesse i.i.d. de l'estimation de la densité est atteinte).

Reçu le : 2006-02-10
Accepté le : 2006-10-10
Publié le : 2006-11-14

DOI : 10.1016/j.crma.2006.10.015

Affiliations des auteurs :

Boris Labrador ¹

¹ L.S.T.A., université Paris 6, 175, rue du Chevaleret, 75013 Paris, France

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Boris Labrador. Almost sure convergence of the $ {k}_{T}$-occupation time density estimator. Comptes Rendus. Mathématique, Volume 343 (2006) no. 10, pp. 665-669. doi : 10.1016/j.crma.2006.10.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.10.015/

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P. Llop; L. Forzani; R. Fraiman On local times, density estimation and supervised classification from functional data, Journal of Multivariate Analysis, Volume 102 (2011) no. 1, p. 73 | DOI:10.1016/j.jmva.2010.08.002
Boris Labrador Strong pointwise consistency of the kT -occupation time density estimator, Statistics Probability Letters, Volume 78 (2008) no. 9, p. 1128 | DOI:10.1016/j.spl.2007.11.010

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