The Esscher Approximation

Abstract

Since it is known that the normal distribution does not give a good approximation if the basic distribution S(z) is very heterogeneous, particularly if at the same time the expected number of claims n is small, another approximation found by Esscher (1932) has been widely used. Unfortunately the error of Esscher’s formula also remains a feature which is difficult to handle mathematically, but even so it has been found in practice to have a broad domain of applicability. The new methods recently developed, i.e. NP-approximation, Section 4.3, Monte Carlo method, Chapter 7, an inversion method, Section 8.1, may provide arguments for re-evaluation of the expediency of the Esscher formula, but nevertheless this method is one of the important tools available for computing the numerical values of the generalized Poisson function and it will now be introduced. A reader who is not familiar with the manipulation of integral transformations and series can at the first reading jump directly to the result (6.7) on page 79.