mgcv (version 1.5-1)

Tweedie: 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.

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.

Details

A Tweedie random variable with 1N 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 1ldTweedie 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.

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)