Nonparametric adaptive estimation for pure jump Lévy processes.

Fabienne Comte (1) , Valentine Genon-Catalot (1)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

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.

Mots clés: Nonparametric Projection Estimator; Adaptive Estimation; Deconvolution; Lévy process; Nonparametric Projection Estimator.

Domaines: Statistiques [math.ST]

Fabienne Comte, Valentine Genon-Catalot. Nonparametric adaptive estimation for pure jump Lévy processes.. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2010, 46 (3), pp.595-617. ⟨10.1214/09-AIHP323⟩. ⟨hal-00289364⟩