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Estimation of the Jump Size Density in a Mixed Compound Poisson Process

Author

Listed:
  • Fabienne Comte
  • Celine Duval
  • Valentine Genon-Catalot
  • Johanna Kappus

Abstract

type="main" xml:id="sjos12149-abs-0001"> In this paper, we consider a mixed compound Poisson process, that is, a random sum of independent and identically distributed (i.i.d.) random variables where the number of terms is a Poisson process with random intensity. We study nonparametric estimators of the jump density by specific deconvolution methods. Firstly, assuming that the random intensity has exponential distribution with unknown expectation, we propose two types of estimators based on the observation of an i.i.d. sample. Risks bounds and adaptive procedures are provided. Then, with no assumption on the distribution of the random intensity, we propose two non-parametric estimators of the jump density based on the joint observation of the number of jumps and the random sum of jumps. Risks bounds are provided, leading to unusual rates for one of the two estimators. The methods are implemented and compared via simulations.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:scjsta:v:42:y:2015:i:4:p:1023-1044
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    References listed on IDEAS

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    1. Martin Bøgsted & Susan Pitts, 2010. "Decompounding random sums: a nonparametric approach," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(5), pages 855-872, October.
    2. Antonio, Katrien & Beirlant, Jan, 2007. "Actuarial statistics with generalized linear mixed models," Insurance: Mathematics and Economics, Elsevier, vol. 40(1), pages 58-76, January.
    3. 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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