Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Nonparametric adaptive estimation for pure jump Lévy processes

F. Comte and V. Genon-Catalot

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Abstract

This paper is concerned with nonparametric estimation of the Lévy density of a pure jump Lévy process. The sample path is observed at n discrete instants with fixed sampling interval. We construct a collection of estimators obtained by deconvolution methods and deduced from appropriate estimators of the characteristic function and its first derivative. We obtain a bound for the ${\mathbb{L}}^{2}$-risk, under general assumptions on the model. Then we propose a penalty function that allows to build an adaptive estimator. The risk bound for the adaptive estimator is obtained under additional assumptions on the Lévy density. Examples of models fitting in our framework are described and rates of convergence of the estimator are discussed.

Résumé

Ce travail étudie l’estimation non paramétrique de la densité d’un processus de Lévy de saut pur. Les trajectoires sont observées à n instants discrets de pas fixé. Nous construisons une collection d’estimateurs obtenus par des méthodes de type déconvolution, et s’appuyant sur des estimateurs pertinents de la fonction caractéristique et de ses dérivées. Sous des hypothèses assez générales sur le modèle, nous obtenons une borne pour le risque quadratique intégré. Nous proposons ensuite une pénalité permettant de construire un estimateur adaptatif. La borne de risque de l’estimateur adaptatatif est obtenue sous des hypothèses supplémentaires sur la densité de la mesure de Lévy. Nous donnons pour finir des exemples de modèles adaptés à notre contexte et nous calculons dans chaque cas la vitesse de convergence de l’estimateur.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 46, Number 3 (2010), 595-617.

Dates
First available in Project Euclid: 6 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1281100391

Digital Object Identifier
doi:10.1214/09-AIHP323

Mathematical Reviews number (MathSciNet)
MR2682259

Zentralblatt MATH identifier
1201.62042

Subjects
Primary: 62G05: Estimation 62M05: Markov processes: estimation 60G51: Processes with independent increments; Lévy processes

Keywords
Adaptive estimation Deconvolution Lévy process Nonparametric projection estimator

Citation

Comte, F.; Genon-Catalot, V. Nonparametric adaptive estimation for pure jump Lévy processes. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 3, 595--617. doi:10.1214/09-AIHP323. https://projecteuclid.org/euclid.aihp/1281100391


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