Adaptive kernel estimation of the Lévy density

Mélina Bec (1) , Claire Lacour (2)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

This paper is concerned with adaptive kernel estimation of the Lévy density $n(x)$ for pure jump Lévy processes. The sample path is observed at $n$ discrete instants in the "high frequency" context ($ \Delta $ = $ \Delta_n $ tends to zero while $n \Delta_n $ tends to $\infty$). We construct a collection of kernel estimators of the function $g(x)=xn(x)$ and propose two methods of local adaptive selection of the bandwidth. The quadratic pointwise risk of the adaptive estimators is studied in both cases. The rate of convergence is proved to be optimal up to a logarithmic factor. We give examples and simulation results for processes fitting in our framework.

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Mélina Bec, Claire Lacour. Adaptive kernel estimation of the Lévy density. 2011. ⟨hal-00583221v1⟩