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Tweedie Tweedie generalized linear model family
DESCRIPTION
Produces a generalized linear model family object with any power variance function and any power link. Includes the Gaussian, Poisson, gamma and inverse-Gaussian families as special cases.
 
USAGE
tweedie(var.power = 0, link.power = 1-var.power)
 
OPTIONAL ARGUMENTS
var.power index of power variance function
link.power index of power link function. link.power=0 produces a log-link. Defaults to the canonical link, which is 1-var.power.
VALUE
A family object, which is a list of functions and expressions used by glm and gam in their iteratively reweighted least-squares algorithms. See family.object in the S-Plus help for details.
 
DETAILS
This function provides access to a range of generalized linear model response distributions which are not otherwise provided by S-Plus, or any other package for that matter. It is also useful for accessing distribution/link combinations which are perversely disallowed by S-Plus, such as Inverse-Gaussion/Log or Gamma/Identity.

Let mi = E( yi) be the expectation of the ith response. We assume that

miq = xiTb,    var( yi) = f mip

where xi is a vector of covariates and b is a vector of regression cofficients, for some f, p and q. This family is specified by var.power = p and link.power = q. A value of zero for q is interpreted as log(mi) = xiTb.

The variance power p characterizes the distribution of the responses y. The following are some special cases:

p Response distribution
0 Normal
1 Poisson
(1, 2) Compound Poisson, non-negative with mass at zero
2 Gamma
3 Inverse-Gaussian
> 2 Stable, with support on the positive reals

The name Tweedie has been associated with this family by Jørgensen in honour of M. C. K. Tweedie.

REFERENCES
Tweedie, M. C. K. (1984). An index which distinguishes between some important exponential families. In Statistics: Applications and New Directions. Proceedings of the Indian Statistical Institute Golden Jubilee International Conference. (Eds. J. K. Ghosh and J. Roy), pp. 579-604. Calcutta: Indian Statistical Institute.
 
Jørgensen, B. (1987). Exponential dispersion models. J. R. Statist. Soc. B, 49, 127-162.
 
Smyth, G. K. (1996). Regression modelling of quantity data with exact zeroes. Proceedings of the Second Australia-Japan Workshop on Stochastic Models in Engineering, Technology and Management. Technology Management Centre, University of Queensland, 572-580. (postscript file)
 
Jørgensen, B. (1997). Theory of Dispersion Models, Chapman and Hall, London.
 
Smyth, G. K., and Verbyla, A. P., (1999). Adjusted likelihood methods for modelling dispersion in generalized linear models. Environmetrics 10, 695-709.
 
SEE ALSO
Tweedie Distributions, qres, Poison-gamma Distribution, inverse-Gaussian Distribution
 
EXAMPLES
# Fit a poisson generalized linear model with identity link
glm(y~x,family=tweedie(var.power=1,link.power=1))

# Fit an inverse-Gaussion glm with log-link
glm(y~x,family=tweedie(var.power=3,link.power=0))

 

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Gordon Smyth. Copyright © 1996-2003. Last modified: 24 December 2002