mgcv (version 1.7-13)

ldTweedie: Log Tweedie density evaluation

Description

A function to evaluate the log of the Tweedie density for variance powers between 1 and 2, inclusive. Also evaluates first and second derivatives of log density w.r.t. its scale parameter.

Usage

ldTweedie(y,mu=y,p=1.5,phi=1)

Arguments

y

values at which to evaluate density.

mu

corresponding means (either of same length as y or a single value).

p

the variance of y is proportional to its mean to the power p. p must be between 1 and 2. 1 is Poisson like (exactly Poisson if phi=1), 2 is gamma.

phi

The scale parameter. Variance of y is phi*mu^p.

Value

• A matrix with 3 columns. The first is the log density of y (log probability if p=1). The second and third are the first and second derivatives of the log density w.r.t. phi.

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 1

ldTweedie 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. If you really need 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.

Examples

library(mgcv)
  ## convergence to Poisson illustrated
  ## notice how p>1.1 is OK
  y <- seq(1e-10,10,length=1000)
  p <- c(1.0001,1.001,1.01,1.1,1.2,1.5,1.8,2)
  phi <- .5
  fy <- exp(ldTweedie(y,mu=2,p=p[1],phi=phi)[,1])
  plot(y,fy,type="l",ylim=c(0,3),main="Tweedie density as p changes")
  for (i in 2:length(p)) {
    fy <- exp(ldTweedie(y,mu=2,p=p[i],phi=phi)[,1])
    lines(y,fy,col=i)
  }