0th

Percentile

Saddlepoint density for the Tweedie distributions

Keywords
models
##### Usage
dtweedie.saddle(y, xi=NULL, mu, phi, eps=1/6, power=NULL)
##### Arguments
y

the vector of responses

xi

the value of $\xi$ such that the variance is $\mbox{var}[Y]=\phi\mu^{\xi}$

power

a synonym for $\xi$

mu

the mean

phi

the dispersion

eps

the offset in computing the variance function. The default is eps=1/6 (as suggested by Nelder and Pregibon, 1987).

##### Details

The Tweedie family of distributions belong to the class of exponential dispersion models (EDMs), famous for their role in generalized linear models. The Tweedie distributions are the EDMs with a variance of the form $\mbox{var}[Y]=\phi\mu^p$ where $p$ is greater than or equal to one, or less than or equal to zero. This function only evaluates for $p$ greater than or equal to one. Special cases include the normal ($p=0$), Poisson ($p=1$ with $\phi=1$), gamma ($p=2$) and inverse Gaussian ($p=3$) distributions. For other values of power, the distributions are still defined but cannot be written in closed form, and hence evaluation is very difficult.

When $1<p<2$, the distribution are continuous for $Y$ greater than zero, with a positive mass at $Y=0$. For $p>2$, the distributions are continuous for $Y$ greater than zero.

This function approximates the density using the saddlepoint approximation defined by Nelder and Pregibon (1987).

##### Value

saddlepoint (approximate) density for the given Tweedie distribution with parameters mu, phi and power.

##### References

Daniels, H. E. (1954). Saddlepoint approximations in statistics. Annals of Mathematical Statistics, 25(4), 631--650.

Daniels, H. E. (1980). Exact saddlepoint approximations. Biometrika, 67, 59--63. 10.1093/biomet/67.1.59

Dunn, P. K. and Smyth, G. K. (2008). Evaluation of Tweedie exponential dispersion model densities by Fourier inversion. Statistics and Computing, 18, 73--86. 10.1007/s11222-007-9039-6

Dunn, Peter K and Smyth, Gordon K (2001). Tweedie family densities: methods of evaluation. Proceedings of the 16th International Workshop on Statistical Modelling, Odense, Denmark, 2--6 July

Dunn, Peter K and Smyth, Gordon K (2005). Series evaluation of Tweedie exponential dispersion model densities Statistics and Computing, 15(4). 267--280. 10.1007/s11222-005-4070-y

Jorgensen, B. (1987). Exponential dispersion models. Journal of the Royal Statistical Society, B, 49, 127-162.

Jorgensen, B. (1997). Theory of Dispersion Models, Chapman and Hall, London.

Nelder, J. A. and Pregibon, D. (1987). An extended quasi-likelihood function. Biometrika, 74(2), 221--232. 10.1093/biomet/74.2.221

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.

dtweedie

##### Examples
# NOT RUN {
p <- 2.5
mu <- 1
phi <- 1
y <- seq(0, 10, length=100)
fy <- dtweedie( y=y, power=p, mu=mu, phi=phi)
plot(y, fy, type="l")
# Compare to the saddlepoint density