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.
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
This is a preview of subscription content, log in to check access.
Barndorff-Nielsen O (1978) Information and exponential families in statistical theory. WileyGoogle Scholar
Choudhury GL, Lucantoni D, Whitt W (1994) Multidimensional transform inversion application to the transient M / G / 1 queue. Ann Appl Probab 4(3):719–740MathSciNetCrossRefzbMATHGoogle Scholar
Diaconis P, Griffiths R (2012) Exchangeables pairs of Bernouilli random variables, Krawtchouck polynomials, and Ehrenfest urns. Australian New Zealand J Stat 54(1):81–101MathSciNetCrossRefzbMATHGoogle Scholar
Dowton F (1970) Bivariate exponential distributions in reliabilty theory. J R Stat Soc Ser B Methodol:408–417Google Scholar
Goffard PO (2015) Approximations polynomiales de densités de probabilité et applications en assurance. PhD thesis. Aix-Marseille University, Avenue de LuminyGoogle 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 pressGoogle Scholar