Nonparametric adaptive estimation for pure jump Lévy processes

Details Nonparametric adaptive estimation for pure jump Lévy processes Nonparametric adaptive estimation for pure jump Lévy processes

2008-06-20

arxiv.org/abs/0806.3371v1

Annales de l'Institut Henri Poincare (B) Probability and Statistics 46, 3 (2010) 595-617

Mathematics

Statistics Theory

Scientific paper

10.1214/09-AIHP323

This paper is concerned with nonparametric estimation of the L\'evy density of a pure jump L\'evy 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 L\'evy density. Examples of models fitting in our framework are described and rates of convergence of the estimator are discussed.

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