tweedie {statmod} | R Documentation |

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.

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

`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` . |

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 *μ_i = E(y_i)* be the expectation of the *i*th response. We assume that

*μ_i^q = x_i^Tb, var(y_i) = phi μ_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(μ_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 Jørgensen in honour of M. C. K. Tweedie.

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.

Gordon Smyth

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, pp. 572-580.

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.

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))

[Package *statmod* version 1.2.4 Index]