Risk bounds for the non-parametric estimation of Lévy processes

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 $\mathcal{S}$ for the true Lévy density. The second is a data-driven selection of a linear model $\mathcal{S}$, among a given collection $\{{\mathcal{S}}_m{\}}_{m\in \mathcal{M}}$, that approximately realizes the best trade-off between the error of estimation within $\mathcal{S}$ and the error incurred when approximating the true Lévy density by the linear model $\mathcal{S}$. 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.

Published at http://dx.doi.org/10.1214/074921706000000789 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Dimension free and infinite variance tail estimates on Poisson space

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