Statistical Papers

, 36:49 | Cite as

On approximate inference for the two-parameter gamma model

Articles

Abstract

In applied work, the two-parameter gamma model gives useful representations of many physical situations. It has a two dimensional sufficient statistic for the two parameters which describe shape and scale. This makes it superficially comparable to the normal model, but accurate and simple statistical inference procedures for each parameter have not been available. In this paper, the saddlepoint approximation is applied to approximate observed levels of significance of the shape parameter. An averaging method is proposed to approximate observed levels of significance of the scale parameter. These methods are extended to the two-sample case.

Keywords

Averaging Confidence distribution function Observed level of significance Saddlepoint approximation Stirling’s formula 

References

  1. Bain, L.J. and Engelhardt, M. (1975). A two moment chi-square approximation for the statistic log\(\left( {{{\bar x} \mathord{\left/ {\vphantom {{\bar x} {\tilde x}}} \right. \kern-\nulldelimiterspace} {\tilde x}}} \right)\).Journal of the American Statistical Association 70, 948–950.MATHGoogle Scholar
  2. Barndorff-Nielsen, O.E. (1986). Inference on full or partial parameters based on the standardized signed log likelihood ratio.Biometrika 73, 307–322.MathSciNetMATHGoogle Scholar
  3. Bishop, D.J. and Nair, U.S. (1939). A note on certain methods of testing for the homogeneity of a set of estimated variances.Journal of the Royal Statistical Society Supplement 6, 89–99.CrossRefMATHGoogle Scholar
  4. Cox, D.R. (1975). A note on partially Bayes inference and linear models.Biometrika 62, 651–654.MathSciNetCrossRefMATHGoogle Scholar
  5. Daniels, H.E. (1954). Saddlepoint approximation in statistics.Annals of Mathematical Statistics 25, 631–650.MathSciNetCrossRefMATHGoogle Scholar
  6. Fisher, R.A. (1934). Two new properties of mathematical likelihood.Proceedings of the Royal Society A 144, 285–307.CrossRefMATHGoogle Scholar
  7. Fraser, D.A.S. (1968).The Structure of Inference. New York: McGraw Hill.MATHGoogle Scholar
  8. Fraser, D.A.S. (1991). Statistical inference: likelihood to significance.Journal of the American Statistical Association 86, 258–265.MathSciNetCrossRefMATHGoogle Scholar
  9. Fraser, D.A.S., Reid, N. and Wong, A. (1991). Exponential linear models: a two pass procedure for saddlepoint approximation.Journal of the Royal Statistical Society B 53, 483–492.MathSciNetMATHGoogle Scholar
  10. Fraser, D.A.S. and Wong, A. (1993). Approximate studentization with marginal and conditional inference.Canadian Journal of Statistics 21, 313–320.MathSciNetCrossRefMATHGoogle Scholar
  11. Jensen, J.L. and Kristensen, L.B. (1991). Saddlepoint approximations to exact tests and improved likelihood ratio tests for the gamma distribution.Communication in Statistics: Theory and Methods 20, 1515–1532.MathSciNetCrossRefGoogle Scholar
  12. Johnson, N.L. and Kotz, S. (1970).Continuous Distribution I. New York: Wiley.MATHGoogle Scholar
  13. Kalbfleisch, J.D. and Sprott, D.A. (1973). Marginal and conditional likelihood.Sankyhā A 35, 311–328.MathSciNetMATHGoogle Scholar
  14. Lugannani, R. and Rice, S.O. (1980). Saddlepoint approximation for the distribution of the sum of independent random variables.Advanced Applied Probability 12, 479–490.MathSciNetCrossRefMATHGoogle Scholar
  15. Pierce, D.A. and Peters, D. (1992). Practical use of higher order asymptotics for multiparameter exponential families.Journal of the Royal Statistical Society B 54, 701–725.MathSciNetGoogle Scholar
  16. Reid, N. (1988). Saddlepoint methods and statistical inference.Statistical Science 3 213–238.MathSciNetCrossRefMATHGoogle Scholar
  17. Shuie, W.K. and Bain, L.J. (1983). A two-sample test of equal gamma distribution scale parameters with unknown common shape parameter.Technometrics 25, 377–381.MathSciNetCrossRefMATHGoogle Scholar
  18. Shuie, W.K., Bain, L.J. and Engelhardt, M. (1988). Test of equal gamma distribution means with unknown and unequal shape parameters.Technometrics 30, 169–174.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • A. Wong
    • 1
  1. 1.Department of Mathematics and StatisticsYork UniversityNorth YorkCanada

Personalised recommendations