Poisson approximation by constrained exponential tilting

Abstract

Poisson approximations are important whenever interest is in some counting process. Barbour et al. (Poisson Approximation, Oxford University Press, New York, 1992), Barbour and Jensen (Scand. J. Statist. 16 (1989) 75), and Kathman and Terrell (J. Statist. Plann. Inference 104(2) (2002) 315) have presented methods for improving Poisson approximations using the ideas of tilting and expansion. These methods are similar to the saddlepoint methods presented by Daniels (Internat. Statist. Rev. 55 (1987) 37). This paper will develop simple and accurate approximations by building on the ideas of tilting and expanding. These approximations are then generalized so as to be useful for cases where the tilting parameter, q, is difficult to obtain. An example is presented to show how these methods compare with the methods developed through the usual combination of exponential tilting and expansion.Poisson approximation Probability generating function Exponential tilting Saddlepoint methods