Risk bounds for the non-parametric estimation of Lévy processes

By José E. Figueroa-López, C. Houdr'e. 2006

56 citations recorded by the source corpus.

Estimation methods for the Levy density of a Levy process are developed under mild qualitative assumptions. A classical model selection ap- proach 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 S for the true Levy density. The second is a data-driven selection of a linear model S, among a given collection {Sm}m∈M, that approximately realizes the best trade-off between the error of estimation within S and the error incurred when approximating the true Levy density by the linear model 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.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/2c69a8034b72de17de10289d37b8b715636ece0d

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Citations in the corpus

Nonparametric estimation for L\'evy processes from low-frequency observations (2007, methodology): “The rates should also be compared with the case of continuous-time observations on [0, T ], where Figueroa-López and Houdré (2006) obtained the classical nonparametric rate T−r/(2r+1) for estimating gσ on a bounded interval.”
Near optimal thresholding estimation of a Poisson intensity on the real line (2008, background): “Besides, in [7], [4] and [19], minimax performances on classical functional spaces are derived only for compactly supported signals.”
Nonparametric Estimation for Lévy Models Based on Discrete-Sampling (2009, methodology): “We hope to extend in a future work the adaptability result in [ 16 ] for this discretebased p.p.e.”