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Polynomial Approximations for Bivariate Aggregate Claims Amount Probability Distributions

Methodology and Computing in Applied Probability volume 19, pages 151–174 (2017)Cite this article

Abstract

A numerical method to compute bivariate probability distributions from their Laplace transforms is presented. The method consists in an orthogonal projection of the probability density function with respect to a probability measure that belongs to a Natural Exponential Family with Quadratic Variance Function (NEF-QVF). A particular link to Lancaster probabilities is highlighted. The procedure allows a quick and accurate calculation of probabilities of interest and does not require strong coding skills. Numerical illustrations and comparisons with other methods are provided. This work is motivated by actuarial applications. We aim at recovering the joint distribution of two aggregate claims amounts associated with two insurance policy portfolios that are closely related, and at computing survival functions for reinsurance losses in presence of two non-proportional reinsurance treaties.

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References

  • Abate J, Choudhury GL, Whitt W (1995) On the Laguerre method for numerically inverting Laplace transforms. INFORMS J Comput 8(4):413–427

    Article  MATH  Google Scholar 

  • Abate J, Choudhury GL, Whitt W (1998) Numerical inversion of multidimensional Laplace transform by the Laguerre method. Perform Eval 31(3):229–243

    Article  Google Scholar 

  • Ambagaspitiya RS (1998) On the distribution of a sum of correlated aggregate claims. Insur Math Econ 23(1):15–19

    Article  MathSciNet  MATH  Google Scholar 

  • Ambagaspitiya RS (1999) On the distributions of two classes of correlated aggregate claims. Insur Math Econ 24(3):301–308

    Article  MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen O (1978) Information and exponential families in statistical theory. Wiley

  • Choudhury GL, Lucantoni D, Whitt W (1994) Multidimensional transform inversion application to the transient M / G / 1 queue. Ann Appl Probab 4(3):719–740

    Article  MathSciNet  MATH  Google Scholar 

  • Diaconis P, Griffiths R (2012) Exchangeables pairs of Bernouilli random variables, Krawtchouck polynomials, and Ehrenfest urns. Australian New Zealand J Stat 54(1):81–101

    Article  MathSciNet  MATH  Google Scholar 

  • Dowton F (1970) Bivariate exponential distributions in reliabilty theory. J R Stat Soc Ser B Methodol:408–417

  • Goffard PO (2015) Approximations polynomiales de densités de probabilité et applications en assurance. PhD thesis. Aix-Marseille University, Avenue de Luminy

    Google Scholar 

  • Goffard PO, Loisel S, Pommeret D (2015) A polynomial expansion to approximate the ultimate ruin probability in the compound Poisson ruin model. J Comput Appl Math. In press

  • He Q, Nagaraja HN, Wu C (2013) Efficient simulation of complete and censored samples from common bivariate distributions. Comput Stat 28(6):2479–2494

    Article  MathSciNet  MATH  Google Scholar 

  • Hesselager O (1996) Recursions for certain bivariate counting distributions and their compound distributions. Astin Bull 26(1):35–52

    Article  MathSciNet  Google Scholar 

  • Jin T, Ren J (2010) Recursions and fast fourier transforms for certain bivariate compound distributions. J Oper Risk 4:19

    Article  Google Scholar 

  • Jin T, Ren J (2014) Recursions and fast fourier transforms for a new bivariate aggregate claims model. Scand Actuar J 8:729–752

    Article  MathSciNet  Google Scholar 

  • Koudou AE (1995) Problèmes de marges et familles exponentielles naturelles, PhD thesis, Toulouse

  • Koudou AE (1996) Probabilités de Lancaster. Expo Math 14:247–276

    MathSciNet  MATH  Google Scholar 

  • Koudou AE (1998) Lancaster bivariate probability distributions with Poisson, negative binomial and gamma margins. Test 7(1):95–110

    Article  MathSciNet  MATH  Google Scholar 

  • Lancaster HO (1958) The structure of bivariate distributions. Ann Math Stat:719–736

  • Marshall AW, Olkin L (1967) A multivariate exponential distribution. J Am Stat Assoc 62(317):30–44

    Article  MathSciNet  MATH  Google Scholar 

  • Mnatsakanov RM (2011) Moment-recovered approximations of multivariate distributions: the Laplace transform inversion. Stat Probab Lett 81(1):1–7

    Article  MathSciNet  MATH  Google Scholar 

  • Mnatsakanov RM, Sarkisian K (2013) A note on recovering the distribution from exponential moments. Appl Math Comput 219:8730–8737

    MathSciNet  MATH  Google Scholar 

  • Mnatsakanov RM, Sarkisian K, Hakobyan A (2015) Approximation of the ruin probability using the scaled Laplace transform inversion. Appl Math Comput 268:717–727

    MathSciNet  Google Scholar 

  • Morris CN (1982) Natural exponential families with quadratic variance functions. Annal Math Stat 10(1):65–80

    Article  MathSciNet  MATH  Google Scholar 

  • Sundt B (1999) On multivariate Panjer recursions. Astin Bull 29(1):29–45

    Article  MathSciNet  Google Scholar 

  • Szegö G (1939) Orthogonal Polynomials, volume XXIII. American mathematical society Colloquium publications

  • Vernic R (1999) Recursive evaluation of some bivariate compound distibutions. Astin Bull

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Authors and Affiliations

  1. Institut de Mathematique de Marseille, Aix-Marseille University, Marseille, France

    Pierre-Olivier Goffard & Denys Pommeret

  2. Université de Lyon, Université Claude Bernard Lyon 1, Institut de Science Actuarielle et Financière, 50 avenue Tony Garnier, 69007, Lyon, France

    Stéphane Loisel

Authors
  1. Pierre-Olivier Goffard
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  2. Stéphane Loisel
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  3. Denys Pommeret
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Correspondence to Pierre-Olivier Goffard.

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Goffard, PO., Loisel, S. & Pommeret, D. Polynomial Approximations for Bivariate Aggregate Claims Amount Probability Distributions. Methodol Comput Appl Probab 19, 151–174 (2017). https://doi.org/10.1007/s11009-015-9470-7

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  • DOI: https://doi.org/10.1007/s11009-015-9470-7

Keywords

  • Bivariate aggregate claims model
  • Bivariate distribution
  • Bivariate laplace transform
  • Numerical inversion of laplace transform
  • Natural exponential families with quadratic variance functions
  • Orthogonal polynomials

Mathematics Subject Classification (2010)

  • 60.08
  • 62P05
  • 65C20
  • 33C45