Series evaluation of Tweedie exponential dispersion model densities

Article

Dunn, Peter K. and Smyth, Gordon K.. 2005. "Series evaluation of Tweedie exponential dispersion model densities." Statistics and Computing. 15 (4), pp. 267-280. https://doi.org/10.1007/s11222-005-4070-y
Article Title

Series evaluation of Tweedie exponential dispersion model densities

ERA Journal ID861
Article CategoryArticle
AuthorsDunn, Peter K. (Author) and Smyth, Gordon K. (Author)
Journal TitleStatistics and Computing
Journal Citation15 (4), pp. 267-280
Number of Pages14
Year2005
Place of PublicationGermany
ISSN0960-3174
1573-1375
Digital Object Identifier (DOI)https://doi.org/10.1007/s11222-005-4070-y
Web Address (URL)https://link.springer.com/article/10.1007%2Fs11222-005-4070-y
Abstract

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
quasi-likelihoood.

Keywordslinear exponential family, generalized linear models, power variance function, Poisson distribution, gamma distribution, inverse-Gaussian distribution, compound Poisson distributions, stable distributions, maximum likelihood estimation
ANZSRC Field of Research 2020490509. Statistical theory
Public Notes

Deposited in accordance with the copyright policy of the publisher.
The original publication is available at www.springerlink.com

Byline AffiliationsDepartment of Mathematics and Computing
Walter and Eliza Hall Institute of Medical Research, Australia
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