Recursive Evaluation of Some Bivariate Compound Distributions

Abstract

In this paper we consider compound distributions where the counting distribution is a bivariate distribution with the probability function (Pn\,n 2 ) ni « 2 >o t n a t satisfies a recursion in the formWe present an algorithm for recursive evaluation of the corresponding compound distributions and some examples of distributions in this class.

Recovery Process Model for Two Companies

“…If frequency distribution p n 1 ,n 2 is the member of (29) can be calculated by the Vernic recursion formula (Theorem A.1 and Theorem A.2). Vernic (1999) proves the following two theorems. 328 Y. Itoh Theorem A.1 If (30) is satisfyed, the compound distribution f S 1 S 2 x 1 , x 2 can be evaluated by the following recursion formula,…”

“…The numerical procedure for calculating the probability function of the bivariate compound Poisson distribution is introduced by Vernic (1999), Sundt (2000), and Powojowski et al (2002). Vernic (1999) and Sundt (2000) extend Panjer (1981) recursion formula, which is the method of calculating the probability distribution function of the compound Poisson distribution. Powojowski et al (2002) calculate the summation of compound Poisson processes by the Monte Carlo simulation.…”

“…For calculating it, we use the Vernic recursion formula which is introduced in Vernic (1999), and the Monte Carlo simulation. For the brief explanation of the Vernic recursion formula, see Section A.…”

“…See Partrat (1994) and Vernic (1999), for details on the bivariate compound Poisson distribution and see also Vernic (1999) for details on Vernic recursion formula.…”

Polynomial Approximations for Bivariate Aggregate Claims Amount Probability Distributions

“…The distribution of the vector X = (X 1 , X 2 ) is recovered through a recursion procedure in [4,24], whereas the distribution of the vector Y = (Y 1 , Y 2 ) is studied in [12,24,26]. Recursion based techniques rely on the existence of recurrence relationships between the probabilities of the claim frequencies which limits their use to certain kind of distributions.…”