Tweedie Distributions (saddlepoint approximation)

Description

Saddlepoint density for the Tweedie distributions

Usage

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dtweedie.saddle(y, xi=power, mu, phi, eps=1/6, power=NULL)

Arguments

y

the vector of responses

xi

the value of xi such that the variance is 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 var(Y) = phi * mu^power where power is greater than or equal to one, or less than or equal to zero. This function only evaluates for power greater than or equal to one. Special cases include the normal (power=0), Poisson (power=1 with phi=1), gamma (power=2) and inverse Gaussian (power=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 < power < 2, the distribution are continuous for Y greater than zero, with a positive mass at Y=0. For power > 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.

Author(s)

Peter Dunn (pdunn2@usc.edu.au)

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.

Dunn, P. K. and Smyth, G. K. (2008). Evaluation of Tweedie exponential dispersion model densities by Fourier inversion. Statistics and Computing, 18, 73–86.

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.

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.

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

dtweedie

Examples

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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
f.saddle <- dtweedie.saddle( y=y, power=p, mu=mu, phi=phi)
lines( y, f.saddle, col=2 )