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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*mu ^{power}**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`p`ower. 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 Jørgensen in honour of M. C. K. Tweedie.

**REFERENCES**- Jørgensen, B. (1987). Exponential dispersion models.
*J. R. Statist. Soc. B*,**49**, 127-162. - Jørgensen, 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. (Postscript file) - 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)`

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Gordon Smyth.
Copyright © 1996-2003. *Last modified:
24 December 2002*