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, phi, and p, or w.r.t. rho=log(phi) and theta where p = (a+b*exp(theta))/(1+exp(theta)).

Usage

ldTweedie(y,mu=y,p=1.5,phi=1,rho=NA,theta=NA,a=1.001,b=1.999,all.derivs=FALSE)

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.
rho	optional log scale parameter. Over-rides phi if theta also supplied.
theta	parameter such that p = (a+b*exp(theta))/(1+exp(theta)). Over-rides p if rho also supplied.
a	lower limit parameter (>1) used in definition of p from theta.
b	upper limit parameter (<2) used in definition of p from theta.
all.derivs	if TRUE then derivatives w.r.t. mu are also returned. Only available with rho and phi parameterization.

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.

The rho, theta parameterization is useful for optimization of p and phi, in order to keep p bounded well away from 1 and 2, and phi positive. The derivatives near p=1 tend to infinity.

Note that if p and phi (or theta and rho) both contain only a single unique value, then the underlying code is able to use buffering to avoid repeated calls to expensive log gamma, di-gamma and tri-gamma functions (mu can still be a vector of different values). This is much faster than is possible when these parameters are vectors with different values.

Value

A matrix with 6 columns, or 10 if all.derivs=TRUE. 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. 4th and 5th columns are first and second derivative w.r.t. p, final column is second derivative w.r.t. phi and p.

If rho and theta were supplied then derivatives are w.r.t. these. In this case, and if all.derivs=TRUE then the 7th colmn is the derivative w.r.t. mu, the 8th is the 2nd derivative w.r.t. mu, the 9th is the mixed derivative w.r.t. theta andmu and the 10th is the mixed derivative w.r.t. rho and mu.

Author(s)

Simon N. Wood [email protected]

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)
  }