# tweedie

##### Tweedie Generalized Linear Models

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.

- Keywords
- regression

##### Usage

`tweedie(var.power=0, link.power=1-var.power)`

##### 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`

.

##### Details

This function provides access to a range of generalized linear model response distributions which are not otherwise provided by R, or any other package for that matter. It is also useful for accessing distribution/link combinations which are disallowed by the R `glm`

function.
Let $\mu_i = E(y_i)$ be the expectation of the $i$th response. We assume that
$$\mu_i^q = x_i^Tb, var(y_i) = \phi \mu_i^p$$
where $x_i$ is a vector of covariates and b is a vector of regression cofficients, for some $\phi$, $p$ and $q$. This family is specified by `var.power = p`

and `link.power = q`

. A value of zero for $q$ is interpreted as $\log(\mu_i) = x_i^Tb$.
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.

##### 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`

and`glm`

in the R base help for details.

##### 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, pp. 572-580.
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

##### Examples

```
y <- rgamma(20,shape=5)
x <- 1:20
# 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))
```

*Documentation reproduced from package statmod, version 1.2.4, License: LGPL version 2 or newer*