On the Calculation of the Generalised Poisson Function

Abstract

Drs. H. Bohman and F. Esscher have reported in a recent paper) an extensive research performed in Sweden on the different methods of calculation of the distribution function of the total amount of claims. In the present paper certain methods are discussed in so far as they are different from those presented in the above quoted paper. The consideration is restricted to the generalised Poisson function even though some results can be easily extended. The author has already commented on some of the results represented in the sequel at a special meeting of the 17th International Congress of Actuaries in Edinburgh.I. Lemma. Let be the generalised Poisson function under investigation. If aiSi(x), where Σ ai = 1 (the functions Si need not be distribution functions, neither must the constants ai be real numbers of interval [0,1]), thenF(x; n, S) = F(.; a1n, S1) * …*F(.; arn, Sr) (x),as is easily verified by the use of characteristic functions. This component representation is repeatedly used in the sequel.2. A Modified Esscher Method. The Esscher method is based on an observation that the well-known Edgeworth expansion is more advantageously applicable to a conveniently modified distribution function instead of the original generalised Poisson function. Let us assume that the value of F(x) is required at a point