Series evaluation of Tweedie exponential dispersion model densities

Dunn, Peter K. and Smyth, Gordon K. (2005) Series evaluation of Tweedie exponential dispersion model densities. Statistics and Computing, 15 (4). pp. 267-280. ISSN 0960-3174


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Exponential dispersion models, which are linear exponential families with a dispersion parameter, are the prototype response distributions for generalized linear models. The Tweedie family comprises those exponential dispersion models with power mean-variance relationships. The normal, Poisson, gamma and inverse Gaussian distributions belong to the Tweedie family. Apart from these special cases, Tweedie distributions do not have density functions which can be written in closed form.
Instead, the densities can be represented as infinite summations derived from series expansions. This
article describes how the series expansions can be summed in an numerically efficient fashion. The usefulness of the approach is demonstrated, but full machine accuracy is shown not to be obtainable using the series expansion method for all parameter values. Derivatives of the density with respect to the dispersion parameter are also derived to facilitate maximum likelihood estimation. The methods are
demonstrated on two data examples and compared with with Box-Cox transformations and extended

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Item Type: Article (Commonwealth Reporting Category C)
Refereed: Yes
Item Status: Live Archive
Additional Information: Deposited in accordance with the copyright policy of the publisher. The original publication is available at
Faculty / Department / School: Historic - Faculty of Sciences - Department of Maths and Computing
Date Deposited: 11 Oct 2007 01:03
Last Modified: 02 Jul 2013 22:43
Uncontrolled Keywords: linear exponential family, generalized linear models, power variance function, Poisson distribution, gamma distribution, inverse-Gaussian distribution, compound Poisson distributions, stable distributions, maximum likelihood estimation
Fields of Research : 01 Mathematical Sciences > 0104 Statistics > 010405 Statistical Theory
Identification Number or DOI: 10.1007/s11222-005-4070-y

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