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Finite Sum Evaluation of the Negative Binomial-Exponential Model

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  • Panjer, Harry H.
  • Willmot, Gordon E.

Abstract

The compound negative binomial distribution with exponential claim amounts (severity) distribution is shown to be equivalent to a compound binomial distribution with exponential claim amounts (severity) with a different parameter. As a result of this, the distribution function and net stop-loss premiums for the Negative Binomial-Exponential model can be calculated exactly as finite sums if the negative binomial parameter α is a positive integer.The result is a generalization of Lundberg (1940).Consider the distribution ofwhere X1, X2, X3, … are independently and identically distributed random variables with common exponential distribution functionand N is an integer valued random variable with probability functionThen the distribution function of S is given byIf MX(t), MN(t) and MS(t) are the associated moment generating functions, then

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  • Panjer, Harry H. & Willmot, Gordon E., 1981. "Finite Sum Evaluation of the Negative Binomial-Exponential Model," ASTIN Bulletin, Cambridge University Press, vol. 12(2), pages 133-137, December.
    • Handle: RePEc:cup:astinb:v:12:y:1981:i:02:p:133-137_00
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    Cited by:

    1. Sarabia, José María & Guillén, Montserrat, 2008. "Joint modelling of the total amount and the number of claims by conditionals," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 466-473, December.
    2. Martel-Escobar, M. & Hernández-Bastida, A. & Vázquez-Polo, F.J., 2012. "On the independence between risk profiles in the compound collective risk actuarial model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(8), pages 1419-1431.
    3. Gómez–Déniz, E. & Pérez–Rodríguez, J.V., 2019. "Modelling distribution of aggregate expenditure on tourism," Economic Modelling, Elsevier, vol. 78(C), pages 293-308.
    4. Nicola Cufaro Petroni & Piergiacomo Sabino, 2019. "Fast Pricing of Energy Derivatives with Mean-reverting Jump-diffusion Processes," Papers 1908.03137, arXiv.org, revised Mar 2020.
    5. Nicola Cufaro Petroni & Piergiacomo Sabino, 2020. "Gamma Related Ornstein-Uhlenbeck Processes and their Simulation," Papers 2003.08810, arXiv.org.

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