R: GAM Tweedie family

Tweedie {mgcv} | R Documentation

## GAM Tweedie family

### Description

A Tweedie family, designed for use with `gam` from the `mgcv` library. Restricted to variance function powers between 1 and 2. A useful alternative to `quasi` when a full likelihood is desirable.

### Usage

    Tweedie(p=1, link = power(0))

### Arguments

`p` | the variance of an observation is proportional to its mean to the power `p`. `p` must be greater than 1 and less than or equal to 2. 1 would be Poisson, 2 is gamma.
`link` | The link function: one of `"log"`, `"identity"`, `"inverse"`, `"sqrt"`, or a `power` link.

### Details

A Tweedie random variable with 1<p<2 is a sum of `N` gamma random variables where `N` has a Poisson distribution. The p=1 case is a generalization of a Poisson distribution and is a discrete distribution supported on integer multiples of the scale parameter. For 1<p<2 the distribution is supported on the positive reals with a point mass at zero. p=2 is a gamma distribution. As p gets very close to 1 the continuous distribution begins to converge on the discretely supported limit at p=1, and is therefore highly multimodal. See `ldTweedie` for more on this behaviour.

`Tweedie` is based partly on the `poisson` family, and partly on `tweedie` from the `statmod` package. It includes extra components to work with all `mgcv` GAM fitting methods as well as an `aic` function. The required log density evaluation (+ derivatives w.r.t. scale) is based on the series evaluation method of Dunn and Smyth (2005).

Without the restriction on `p` the calculation of Tweedie densities is less straightforward, and there does not currently seem to be an implementation which offers any benefit over `quasi`. If you really need to this case then the `tweedie` package is the place to start.

### Value

An object inheriting from class `family`, with additional elements

`dvar` | the function giving the first derivative of the variance function w.r.t. `mu`.
`d2var` | the function giving the second derivative of the variance function w.r.t. `mu`.
`ls` | A function returning a 3 element array: the saturated log likelihood followed by its first 2 derivatives w.r.t. the scale parameter.

### Author(s)

Simon N. Wood simon.wood@r-project.org modified from Venables and Ripley's `negative.binomial` family.

### References

Dunn, P.K. and G.K. Smith (2005) Series evaluation of Tweedie exponential dispersion model densities. Statistics and Computing 15:267-280

Tweedie, M. C. K. (1984). An index which distinguishes between some important exponential families. 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.

### See Also

`ldTweedie`

### Examples

    library(mgcv)
    set.seed(3)
    n<-400
    ## Simulate data (really Poisson with log-link)
    dat <- gamSim(1,n=n,dist="poisson",scale=.2)

    ## Fit a `nearby' Tweedie...
    b <- gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=Tweedie(1.1,power(.1)),
             data=dat)
    plot(b,pages=1)
    print(b)

    ## Same by approximate REML...
    b1 <- gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=Tweedie(1.1,power(.1)),
              data=dat,method="REML")
    plot(b1,pages=1)
    print(b1)

    rm(dat)

[Package mgcv version 1.7-18 Index]