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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 m_i = E( y_i) be the expectation of the ith response. We assume that

m_i^q = x_i^Tb,    var( y_i) = f m_i^p

where x_i 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(m_i) = x_i^Tb.

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 J�rgensen 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.

 

J�rgensen, 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. [PDF]

 

J�rgensen, 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))

Gordon Smyth. Copyright © 1996-2016. Last modified: 10 February 2004