The New Variant of Multivariate Generalization of the Generalized Poisson Distribution1
Н. Л. Иванова · 2002
When we consider the construction of the collective risk model the important step is the choice of the model for the process of claims. In classical risk model the homogeneous Poisson process with parameter λ is used. In this model it is assumed that all claims are the same type and only one claim can occur at a time. However, if we have claims of the different types, then we need to consider simultaneously several processes of claims. In this case several claims can occur at the same moment but they are of different types. Finally these processes can be dependent. So we face with the problem of the definition of multivariate Poisson distribution. In our preceding paper ([1]) we have considered the class Fn of multivariate natural exponential families of probability distributions of random vector X = (X1, . . . , Xn), every subvector of which has the distribution from the analogous family. Let the marginal distributions of this vector be the Poisson ones with parameters λk, k = 1, n. Then the joint distribution of random vector X is defined uniquely and has the following representation. For every collection (i1, . . . , in), ik = 0∨ 1, k = 1, n, letNi1,...,in be independent random variables with Poisson distributions whose parameters are λi1,...,in . In what follows if we write (i1, . . . , 1, . . . , in) it means that we consider the index where the corresponding component is equal to 1. In our paper ([1]) we have shown that random vector X has the following representation: X = ∑