# An Improved Saddlepoint Approximation Based on the Negative Binomial Distribution for the General Birth Process

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## Summary

Daniels (*JAP* 1982) gave a saddlepoint approximation to the probabilities of a general birth process. This paper gives an improved approximation which is only slightly more complex than Daniels’ approximation and which has considerably reduced relative error in most cases. The new approximation has the characteristic that it is exact whenever the birth rates can be reordered into a linear increasing sequence.

## Keywords

saddlepoint approximation birth process negative binomial distribution binomial distribution exponential tilting## Notes

### Acknowledgement

Much of the work leading to this paper was completed while the first author was a visiting Associate Professor at the University of Southern Denmark.

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© Physica-Verlag 2002