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VOL. 51 | 2006 Risk bounds for the non-parametric estimation of Lévy processes
José E. Figueroa-López, Christian Houdré

Editor(s) Evarist Giné, Vladimir Koltchinskii, Wenbo Li, Joel Zinn

Abstract

Estimation methods for the Lévy density of a Lévy process are developed under mild qualitative assumptions. A classical model selection approach made up of two steps is studied. The first step consists in the selection of a good estimator, from an approximating (finite-dimensional) linear model $\calS$ for the true Lévy density. The second is a data-driven selection of a linear model $\calS$, among a given collection $\{\calS_{m}\}_{m\in\calM}$, that approximately realizes the best trade-off between the error of estimation within $\calS$ and the error incurred when approximating the true Lévy density by the linear model $\calS$. Using recent concentration inequalities for functionals of Poisson integrals, a bound for the risk of estimation is obtained. As a byproduct, oracle inequalities and long-run asymptotics for spline estimators are derived. Even though the resulting underlying statistics are based on continuous time observations of the process, approximations based on high-frequency discrete-data can be easily devised.

Information

Published: 1 January 2006
First available in Project Euclid: 28 November 2007

zbMATH: 1117.62085

Digital Object Identifier: 10.1214/074921706000000789

Subjects:
Primary: 60G51 , 62G05
Secondary: 60E07 , 62P05

Keywords: Concentration inequalities , estimation of Levy processes , estimation of Poisson processes , Model selection , Oracle inequalities , projection estimation