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 tweedie Tweedie Distributions
DESCRIPTION
Density, cumulative distribution function and quantiles for the Tweedie distributions. Includes the normal, Poison, Poison-gamma and inverse-Gaussian distributions as special cases.

USAGE
dtweedie(x, mu, phi=1, power=1.5)
ptweedie(q, mu, phi=1, power=1.5)
qtweedie(p, mu, phi=1, power=1.5)

REQUIRED ARGUMENTS
 x vector of deviate values. Missing values (NAs) are allowed. p vector of probabilities. Missing values (NAs) are allowed. q vector of quantiles. Missing values (NAs) are allowed. mu vector of (positive) means.

OPTIONAL ARGUMENTS
 phi vector of (positive) dispersion parameters. power power index of variance function. The variance of the distribution is phi*mupower

VALUE
Vector of same length as x, p or q giving the density (dpoisgam), probability (ppoisgam) or quantile (ppoisgam) for the Poisson gamma distribution with mean mu, dispersion phi and index power. Elements of x, p or q that are missing will cause the corresponding elements of the result to be missing.

BACKGROUND

The variance power p characterizes the distribution of x. The following are some special cases:

p Response distribution
0 Normal
1 Poisson
(1, 2) Compound Poisson, non-negative with mass at zero
2 Gamma
3 Inverse-Gaussian
> 2 Stable, with support on the positive reals

The name Tweedie has been associated with this family by Jřrgensen in honour of M. C. K. Tweedie.

REFERENCES
Jřrgensen, B. (1987). Exponential dispersion models. J. R. Statist. Soc. B, 49, 127-162.

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

Smyth, G. K. (1996). Regression modelling of quantity data with exact zeroes. Proceedings of the Second Australia-Japan Workshop on Stochastic Models in Engineering, Technology and Management. Technology Management Centre, University of Queensland, 572-580. [PDF]

Tweedie, M. C. K. (1984). An index which distinguishes between some important exponential families. In 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.

BUGS
For some values of index, the results are based on a simple normal approximation which is very crude. dtweedie and ptweedie are accurate for p=0,[1,2],3. qtweedie is accurate for p=0,1,2,3.

SEE ALSO
Tweedie family, Poison-gamma distribution, inverse-Gaussian distribution
EXAMPLES
y <- c(-1,0,4,5,10000)
d <- dtweedie(y,mu=4,phi=1,power=1.6)
p <- ptweedie(y,mu=4,phi=1,power=1.6)

Gordon Smyth. Copyright © 1996-2016. Last modified: 10 February 2004