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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 m

_{i}= E(*y*) be the expectation of the_{i}*i*th response. We assume thatm

=_{i}^{q}**x**_{i}^{T}**b**, var(*y*) = f_{i}_{i}^{p}where

is a vector of covariates and**x**_{i}**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}^{T}**b**.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*