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
Abramowitz, M., Stegun, I.A. (eds.): A Handbook of Mathematical Functions. Dover, New York (1965)
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)
Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. Academic, New York (1975)
Dunn, P.K.: Likelihood-based inference for Tweedie exponential dispersion models. Unpublished Ph.D. Thesis, University of Queensland (2001)
Dunn, P.K.: Occurrence and quantity of precipitation can be modelled simultaneously. Int. J. Clim. 24, 1231–1239 (2004)
Dunn, P.K., Smyth, G.K.: Series evaluation of Tweedie exponential dispersion models densities. Stat. Comput. 15, 267–280 (2005)
Evans, G.: Practical Numerical Integration. Wiley, New York (1993)
Feller, W.: An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. Wiley, New York (1971)
Gilchrist, R.: Regression models for data with a non-zero probability of a zero response. Commun. Stat. Theory Methods 29, 1987–2003 (2000)
Johnson, N.L., Kotz, S.: Continuous Univariate Distributions—I. Houghton Mifflin, Boston (1970)
Jørgensen, B.: Exponential dispersion models (with discussion). J. R. Stat. Soc. B 49, 127–162 (1987)
Jørgensen, B.: The Theory of Dispersion Models. Chapman and Hall, London (1997)
Jørgensen, B., Paes de Souza, M.C.: Fitting Tweedie’s compound Poisson model to insurance claims data. Scand. Actuar. J. 1, 69–93 (1994)
Krommer, A.R., Überhuber, C.W.: Computational Integration. Society for Industrial and Applied Mathematics, Philadelphia (1998)
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)
McCullagh, P., Nelder, J.A.: Generalized Linear Models, 2nd edn. Chapman and Hall, London (1989)
Nolan, J.P.: An algorithm for evaluating stable densities in Zolotarev’s (M) parameterization. Math. Comput. Model. 29, 229–233 (1997)
Piessens, R., de Doncker-Kapenga, E., Überhuber, C.W., Kahaner, D.K.: Quadpack: A Subroutine Package for Automatic Integration. Springer, Berlin (1983)
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)
R Development Core Team: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2005). http://www.R-project.org
Rabinowitz, P.: Extrapolation methods in numerical integration. Numer. Algorithms 3, 17–28 (1992)
Seigel, A.F.: The noncentral chi-squared distribution with zero degrees of freedom and testing for uniformity. Biometrika 66, 381–386 (1979)
Seigel, A.F.: Modelling data containing exact zeros using zero degrees of freedom. J. R. Stat. Soc. B 47, 267–271 (1985)
Sidi, A.: Extrapolation methods for oscillatory infinite integrals. IMA J. Appl. Math. 26, 1–20 (1980)
Sidi, A.: An algorithm for a special case of a generalization of the Richardson extrapolation process. Numer. Math. 38, 299–307 (1982a)
Sidi, A.: The numerical evaluation of very oscillatory infinite integrals by extrapolation. Math. Comput. 538, 517–529 (1982b)
Sidi, A.: A user-friendly extrapolation method for oscillatory infinite integrals. Math. Comput. 51, 249–266 (1988)
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)
Sidi, A.: Practical Extrapolation Methods: Theory and Applications. Cambridge University Press, Cambridge (2003)
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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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DOI: https://doi.org/10.1007/s11222-007-9039-6