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Parameter estimation of a bivariate compound Poisson process

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  • Esmaeili, Habib
  • Klüppelberg, Claudia

Abstract

In this article, we review the concept of a Lévy copula to describe the dependence structure of a bivariate compound Poisson process. In this first statistical approach we consider a parametric model for the Lévy copula and estimate the parameters of the full dependent model based on a maximum likelihood approach. This approach ensures that the estimated model remains in the class of multivariate compound Poisson processes. A simulation study investigates the small sample behaviour of the MLEs, where we also suggest a new simulation algorithm. Finally, we apply our method to Danish fire insurance data.

Suggested Citation

  • Esmaeili, Habib & Klüppelberg, Claudia, 2010. "Parameter estimation of a bivariate compound Poisson process," Insurance: Mathematics and Economics, Elsevier, vol. 47(2), pages 224-233, October.
    • Handle: RePEc:eee:insuma:v:47:y:2010:i:2:p:224-233
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    References listed on IDEAS

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    1. Kallsen, Jan & Tankov, Peter, 2006. "Characterization of dependence of multidimensional Lévy processes using Lévy copulas," Journal of Multivariate Analysis, Elsevier, vol. 97(7), pages 1551-1572, August.
    2. Irmingard Eder & Claudia Kluppelberg, 2009. "The first passage event for sums of dependent L\'evy processes with applications to insurance risk," Papers 0912.1925, arXiv.org.
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    1. Habib Esmaeili & Claudia Klüppelberg, 2013. "Two-Step Estimation Of A Multi-Variate Lévy Process," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(6), pages 668-690, November.
    2. J. L. van Velsen, 2012. "Parameter estimation of a Levy copula of a discretely observed bivariate compound Poisson process with an application to operational risk modelling," Papers 1212.0092, arXiv.org.
    3. Esmaeili, Habib & Klüppelberg, Claudia, 2011. "Parametric estimation of a bivariate stable Lévy process," Journal of Multivariate Analysis, Elsevier, vol. 102(5), pages 918-930, May.
    4. Riva-Palacio, Alan & Leisen, Fabrizio, 2021. "Compound vectors of subordinators and their associated positive Lévy copulas," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    5. Oliver Grothe & Stephan Nicklas, 2012. "Vine Constructions of Levy Copulas," Papers 1207.4309, arXiv.org, revised Sep 2012.
    6. Grothe, Oliver & Nicklas, Stephan, 2013. "Vine constructions of Lévy copulas," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 1-15.
    7. Rocco Roberto Cerchiara & Francesco Acri, 2020. "Estimating the Volatility of Non-Life Premium Risk Under Solvency II: Discussion of Danish Fire Insurance Data," Risks, MDPI, vol. 8(3), pages 1-19, July.
    8. Rocco Roberto Cerchiara & Francesco Acri, 2016. "Aggregate Loss Distribution And Dependence: Composite Models, Copula Functions And Fast Fourier Transform For The Danish Re Insurance Data," Working Papers 201608, Università della Calabria, Dipartimento di Economia, Statistica e Finanza "Giovanni Anania" - DESF.

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