SOLVING THE STEIN EQUATION IN COMPOUND POISSON APPROXIMATION

The accuracy of compound Poisson approximation can be estimated using Stein's method in terms of quantities similar to those which must be c alculated for Poisson approximation. However, the solutions of the rel evant Stein equation may, in general, grow exponentially fast with the mean number of 'clumps', leading to many applications in which the bo unds are of little use. In this paper, we introduce a method for circu mventing this difficulty. We establish good bounds for those solutions of the Stein equation which are needed to measure the accuracy of app roximation with respect to Kolmogorov distance, but only in a restrict ed range of the argument. The restriction on the range is then compens ated by a truncation argument. Examples are given to show that the met hod clearly outperforms its competitors, as soon as the mean number of clumps is even moderately large.