Risk bounds for the non-parametric estimation of Lévy processes

Abstract

<!-- *** Custom HTML *** --> 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.

Cite this work

No public copy is stored for this record, so only the metadata is cited.

Stored only when both papers are in the corpus — 44 citations exist upstream.

Papers that cite the same references — the relational fallback while embeddings are still backfilling.