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An oracle inequality for penalised projection estimation of Lévy densities from high-frequency observations

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  • Florian Ueltzhöfer
  • Claudia Klüppelberg

Abstract

We consider a multivariate Lévy process given by the sum of a Brownian motion with drift and an independent time-homogeneous pure jump process governed by a Lévy density. We assume that observation of a sample path takes place on an equidistant discrete time grid. Following Grenander's method of sieves, we construct families of nonparametric projection estimators for the restriction of a Lévy density to bounded sets away from the origin. Moreover, we introduce a data-driven penalisation criterion to select an estimator within a given family, where we measure the estimation error in an L2-norm. Furthermore, we give sufficient conditions on the penalty such that an oracle inequality holds. As an application, we prove adaptiveness for sufficiently smooth Lévy densities in some Sobolev space and explicitly derive the rate of convergence.

Suggested Citation

  • Florian Ueltzhöfer & Claudia Klüppelberg, 2011. "An oracle inequality for penalised projection estimation of Lévy densities from high-frequency observations," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(4), pages 967-989.
    • Handle: RePEc:taf:gnstxx:v:23:y:2011:i:4:p:967-989
    DOI: 10.1080/10485252.2011.581375
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    Cited by:

    1. Mélina Bec & Claire Lacour, 2015. "Adaptive pointwise estimation for pure jump Lévy processes," Statistical Inference for Stochastic Processes, Springer, vol. 18(3), pages 229-256, October.
    2. Mark Anthony Caruana, 2017. "Estimation of Lévy Processes via Stochastic Programming and Kalman Filtering," Methodology and Computing in Applied Probability, Springer, vol. 19(4), pages 1211-1225, December.
    3. Akakpo, Nathalie, 2017. "Multivariate intensity estimation via hyperbolic wavelet selection," Journal of Multivariate Analysis, Elsevier, vol. 161(C), pages 32-57.
    4. Fabienne Comte & Céline Duval & Valentine Genon-Catalot, 2014. "Nonparametric density estimation in compound Poisson processes using convolution power estimators," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(1), pages 163-183, January.

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