# tweedie

0th

Percentile

##### 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
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: cl{ 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 Joergensen (1987) 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. Joergensen, 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. Joergensen, 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.

glm, family, dtweedie

• tweedie
##### 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=3,link.power=0))