ldTweedie {mgcv}R Documentation

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.

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.

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.

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.

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.

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)
  }
[Package mgcv version 1.7-19 Index]