Series evaluation of Tweedie exponential dispersion model densities

Peter K Dunn; G K Smyth

doi:10.1007/s11222-005-4070-y

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Series evaluation of Tweedie exponential dispersion model densities

Journal article

Peer reviewed

Series evaluation of Tweedie exponential dispersion model densities

Peter K Dunn and G K Smyth

Statistics and Computing, Vol.15(4), pp.267-280

2005

DOI: https://doi.org/10.1007/s11222-005-4070-y

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Abstract

Statistics

Computation Theory and Mathematics

linear exponential family

generalized linear models

power variance function

Poisson distribution

gamma distribution

inverse-Gaussian distribution

compound Poisson distribution

stable distributions

maximum likelihood estimation

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 theTweedie 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.

Details

Title: Series evaluation of Tweedie exponential dispersion model densities
Authors: Peter K Dunn (Author) - University of Southern Queensland
G K Smyth (Author) - Walter and Eliza Hall Institute of Medical Research
Publication details: Statistics and Computing, Vol.15(4), pp.267-280
Publisher: Springer New York LLC
Date published: 2005
DOI: 10.1007/s11222-005-4070-y
ISSN: 0960-3174
Organisation Unit: University of the Sunshine Coast, Queensland; School of Health and Sport Sciences - Legacy; School of Science, Technology and Engineering
Language: English
Record Identifier: 99449601902621
Output Type: Journal article

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Web Of Science research areas: Computer Science, Theory & Methods; Statistics & Probability

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