Nonparametric inference for discretely sampled Lévy processes

Nonparametric inference for discretely sampled Lévy processes

Shota Gugushvili

Source: Ann. Inst. H. Poincaré Probab. Statist. Volume 48, Number 1 (2012), 282-307.

Abstract

Given a sample from a discretely observed Lévy process X = (X_t)_t≥0 of the finite jump activity, the problem of nonparametric estimation of the Lévy density ρ corresponding to the process X is studied. An estimator of ρ is proposed that is based on a suitable inversion of the Lévy–Khintchine formula and a plug-in device. The main results of the paper deal with upper risk bounds for estimation of ρ over suitable classes of Lévy triplets. The corresponding lower bounds are also discussed.

Résumé

Soit un échantillon d’un processus de Lévy X = (X_t)_t≥0 à activité finie observé en temps discret, le problème d’estimation non-paramétrique de la densité de Lévy ρ est étudié. Un estimateur de ρ est proposé basé sur une inversion de Fourier de la formule de Lévy–Khintchine et un principe de plug-in. Les principaux résultats de cet article portent sur la majoration du risque de l’estimateur de ρ pour des classes de triplets de Lévy. La minoration du risque est aussi discutée.

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Primary Subjects: 62G07, 62G20

Keywords: Empirical characteristic function; Empirical process; Fourier inversion; Lévy density; Lévy process; Maximal inequality; Mean square error

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