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Nonparametric density estimation in compound Poisson processes using convolution power estimators

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  • Fabienne Comte
  • Céline Duval
  • Valentine Genon-Catalot

Abstract

Consider a compound Poisson process which is discretely observed with sampling interval $$\Delta $$ Δ until exactly $$n$$ n nonzero increments are obtained. The jump density and the intensity of the Poisson process are unknown. In this paper, we build and study parametric estimators of appropriate functions of the intensity, and an adaptive nonparametric estimator of the jump size density. The latter estimation method relies on nonparametric estimators of $$m$$ m th convolution powers density. The $$L^2$$ L 2 -risk of the adaptive estimator achieves the optimal rate in the minimax sense over Sobolev balls. Numerical simulation results on various jump densities enlight the good performances of the proposed estimator. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metrik:v:77:y:2014:i:1:p:163-183
    DOI: 10.1007/s00184-013-0475-3
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    References listed on IDEAS

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    1. Scalas, Enrico, 2006. "The application of continuous-time random walks in finance and economics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 362(2), pages 225-239.
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    4. Chen, Song X. & Delaigle, Aurore & Hall, Peter, 2010. "Nonparametric estimation for a class of Lévy processes," Journal of Econometrics, Elsevier, vol. 157(2), pages 257-271, August.
    5. 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.
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    Cited by:

    1. Blanke, D. & Bosq, D., 2016. "Detecting and estimating intensity of jumps for discretely observed ARMAD(1,1) processes," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 119-137.
    2. Fabienne Comte & Celine Duval & Valentine Genon-Catalot & Johanna Kappus, 2015. "Estimation of the Jump Size Density in a Mixed Compound Poisson Process," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(4), pages 1023-1044, December.
    3. Shota Gugushvili & Frank Meulen & Peter Spreij, 2018. "A non-parametric Bayesian approach to decompounding from high frequency data," Statistical Inference for Stochastic Processes, Springer, vol. 21(1), pages 53-79, April.

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