Evaluation of Tweedie exponential dispersion model densities by Fourier inversion

Statistics and Computing volume 18, pages73–86(2008)Cite this article

Abstract

The Tweedie family of distributions is a family of exponential dispersion models with power variance functions V(μ)=μ p for \(p\not\in(0,1)\) . These distributions do not generally have density functions that can be written in closed form. However, they have simple moment generating functions, so the densities can be evaluated numerically by Fourier inversion of the characteristic functions. This paper develops numerical methods to make this inversion fast and accurate. Acceleration techniques are used to handle oscillating integrands. A range of analytic results are used to ensure convergent computations and to reduce the complexity of the parameter space. The Fourier inversion method is compared to a series evaluation method and the two methods are found to be complementary in that they perform well in different regions of the parameter space.

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References

  1. Abramowitz, M., Stegun, I.A. (eds.): A Handbook of Mathematical Functions. Dover, New York (1965)

    Google Scholar 

  2. Candy, S.G.: Modelling catch and effort data using generalized linear models, the Tweedie distribution, random vessel effects and random stratum-by-year effects. CCAMLR Sci. 11, 59–80 (2004)

    MathSciNet  Google Scholar 

  3. Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. Academic, New York (1975)

    Google Scholar 

  4. Dunn, P.K.: Likelihood-based inference for Tweedie exponential dispersion models. Unpublished Ph.D. Thesis, University of Queensland (2001)

  5. Dunn, P.K.: Occurrence and quantity of precipitation can be modelled simultaneously. Int. J. Clim. 24, 1231–1239 (2004)

    Article  Google Scholar 

  6. Dunn, P.K., Smyth, G.K.: Series evaluation of Tweedie exponential dispersion models densities. Stat. Comput. 15, 267–280 (2005)

    Article  MathSciNet  Google Scholar 

  7. Evans, G.: Practical Numerical Integration. Wiley, New York (1993)

    Google Scholar 

  8. Feller, W.: An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. Wiley, New York (1971)

    Google Scholar 

  9. Gilchrist, R.: Regression models for data with a non-zero probability of a zero response. Commun. Stat. Theory Methods 29, 1987–2003 (2000)

    MATH  Article  Google Scholar 

  10. Johnson, N.L., Kotz, S.: Continuous Univariate Distributions—I. Houghton Mifflin, Boston (1970)

    Google Scholar 

  11. Jørgensen, B.: Exponential dispersion models (with discussion). J. R. Stat. Soc. B 49, 127–162 (1987)

    MATH  Google Scholar 

  12. Jørgensen, B.: The Theory of Dispersion Models. Chapman and Hall, London (1997)

    Google Scholar 

  13. Jørgensen, B., Paes de Souza, M.C.: Fitting Tweedie’s compound Poisson model to insurance claims data. Scand. Actuar. J. 1, 69–93 (1994)

    Google Scholar 

  14. Krommer, A.R., Überhuber, C.W.: Computational Integration. Society for Industrial and Applied Mathematics, Philadelphia (1998)

    Google Scholar 

  15. Lambert, P., Lindsey, J.K.: Analysing financial returns using regression models based on non-symmetric stable distributions. J. R. Stat. Soc. C 48, 409–424 (1999)

    MATH  Article  Google Scholar 

  16. McCullagh, P., Nelder, J.A.: Generalized Linear Models, 2nd edn. Chapman and Hall, London (1989)

    Google Scholar 

  17. Nolan, J.P.: An algorithm for evaluating stable densities in Zolotarev’s (M) parameterization. Math. Comput. Model. 29, 229–233 (1997)

    Article  Google Scholar 

  18. Piessens, R., de Doncker-Kapenga, E., Überhuber, C.W., Kahaner, D.K.: Quadpack: A Subroutine Package for Automatic Integration. Springer, Berlin (1983)

    Google Scholar 

  19. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in FORTRAN 77: The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  20. R Development Core Team: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2005). http://www.R-project.org

    Google Scholar 

  21. Rabinowitz, P.: Extrapolation methods in numerical integration. Numer. Algorithms 3, 17–28 (1992)

    MATH  Article  MathSciNet  Google Scholar 

  22. Seigel, A.F.: The noncentral chi-squared distribution with zero degrees of freedom and testing for uniformity. Biometrika 66, 381–386 (1979)

    Article  MathSciNet  Google Scholar 

  23. Seigel, A.F.: Modelling data containing exact zeros using zero degrees of freedom. J. R. Stat. Soc. B 47, 267–271 (1985)

    Google Scholar 

  24. Sidi, A.: Extrapolation methods for oscillatory infinite integrals. IMA J. Appl. Math. 26, 1–20 (1980)

    MATH  Article  MathSciNet  Google Scholar 

  25. Sidi, A.: An algorithm for a special case of a generalization of the Richardson extrapolation process. Numer. Math. 38, 299–307 (1982a)

    MATH  Article  MathSciNet  Google Scholar 

  26. Sidi, A.: The numerical evaluation of very oscillatory infinite integrals by extrapolation. Math. Comput. 538, 517–529 (1982b)

    MathSciNet  Google Scholar 

  27. Sidi, A.: A user-friendly extrapolation method for oscillatory infinite integrals. Math. Comput. 51, 249–266 (1988)

    MATH  Article  MathSciNet  Google Scholar 

  28. Sidi, A.: Computation of infinite integrals involving Bessel functions or arbitrary order by the \(\bar{D}\) -transformation. J. Comput. Appl. Math. 78, 125–130 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  29. Sidi, A.: Practical Extrapolation Methods: Theory and Applications. Cambridge University Press, Cambridge (2003)

    Google Scholar 

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Affiliations

  1. Australian Centre for Sustainable Catchments and Department of Mathematics and Computing, University of Southern Queensland, Toowoomba, QLD, 4350, Australia

    Peter K. Dunn

  2. Bioinformatics Division, Walter and Eliza Hall Institute of Medical Research, Melbourne, VIC, 3050, Australia

    Gordon K. Smyth

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  1. Peter K. Dunn
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  2. Gordon K. Smyth
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Correspondence to Peter K. Dunn.

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Dunn, P.K., Smyth, G.K. Evaluation of Tweedie exponential dispersion model densities by Fourier inversion. Stat Comput 18, 73–86 (2008). https://doi.org/10.1007/s11222-007-9039-6

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Keywords

  • Compound Poisson distribution
  • Generalized linear models
  • Numerical integration
  • Numerical acceleration
  • Power variance function