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Computational Statistics

, Volume 17, Issue 1, pp 17–28 | Cite as

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

  • Gordon K. Smyth
  • Heather M. Podlich
Article

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.

References

  1. Daniels, H. E. (1954). Saddlepoint approximations in statistics. Annals of Mathematical Statistics 25, 631–650.MathSciNetCrossRefGoogle Scholar
  2. Daniels, H. E. (1982). The saddlepoint approximation for a general birth process. Journal of Applied Probability, 19, 20–28.MathSciNetCrossRefGoogle Scholar
  3. Embrechts, P., Jensen, J. L., Maejima, M., and Teugels, J. L. (1985). Approximations for compound Poisson and Polya processes. Advances of Applied Probability, 17, 623–637.CrossRefGoogle Scholar
  4. Faddy, M. J. (1997a). Extended Poisson process modelling and analysis of count data. Biometrical Journal 39, 431–440.CrossRefGoogle Scholar
  5. Faddy, M. J. (1997b). On extending the negative binomial distribution, and the number of weekly winners of the UK national lottery. Math. Scientist. 22, 77–82.MathSciNetzbMATHGoogle Scholar
  6. Faddy, M. J. and Bosch, R.J. (1999). Likelihood based modelling and analysis of data under-dispersed relative to the Poisson distribution. Biometrics. To appear.Google Scholar
  7. Grassman, W. (1977). Transient solutions in Markovian queuing systmes. Computers and Operations Research, 4, 47–66.CrossRefGoogle Scholar
  8. Jensen, J. L. (1991). Uniform saddlepoint approximations and log-concave densities. Journal Royal Statistical Society B53, 157–172.MathSciNetzbMATHGoogle Scholar
  9. Jensen, J. L. (1991). A large deviation-type approximation for the “Box class” of likelihood ratio criteria. Journal of the American Statistical Society, 86, 437–440.MathSciNetGoogle Scholar
  10. Jensen, J. L. (1995). Saddlepoint Approximations. Oxford University Press, New York.zbMATHGoogle Scholar
  11. McCullagh, P. (1987). Tensor Methods in Statistics. Chapman and Hall, London.zbMATHGoogle Scholar
  12. Podlich, H. M., Faddy, M. J., and Smyth, G. K. (1999a). Likelihood computations for extended Poisson process models. InterStat, September #1.Google Scholar
  13. Podlich, H. M., Faddy, M. J., and Smyth, G. K. (1999b). Semi-parametric extended Poisson process models. Research Report 23, Department of Statistics and Demography, University of Southern Denmark, Odense.Google Scholar
  14. Reid, N. (1988). Saddlepoint methods and statistical inference. With comments and a rejoinder by the author. Statistical Science, 3, 213–238.MathSciNetCrossRefGoogle Scholar
  15. Sidje, R. B. (1998). EXPOKIT software package for computing matrix exponentials. A CM Transactions on Mathematical Software, 24, 130–156.CrossRefGoogle Scholar
  16. Smyth, G. K., and Podlich, H. M. (2000). Score tests for Poisson variation against general alternatives. Computing Science and Statistics 33. To appear.Google Scholar
  17. Stewart, W. J. (1994). Introduction to the Numerical Solutions of Markov chains. Princeton University Press, Princeton, New Jersey.zbMATHGoogle Scholar
  18. Stewart, W. J. (ed.) (1995). Computations with Markov Chains: Proceedings of the 2nd International Workshop on the Numerical Solution of Markov Chains. Kluwer Academic Publishers, Norwell, Massachusetts.zbMATHGoogle Scholar

Copyright information

© Physica-Verlag 2002

Authors and Affiliations

  • Gordon K. Smyth
    • 1
  • Heather M. Podlich
    • 1
  1. 1.Department of Mathematics and School of Land and FoodUniversity of QueenslandBrisbaneAustralia

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