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poisgam |
Poisson Gamma
Distribution |

**DESCRIPTION**- Density, cumulative probability function and random variates for the Poisson-gamma or compound Poisson distribution. The Poison-gamma distribution is a Tweedie distribution with index p between 1 and 2.
**USAGE**`dpoisgam(x, mu, phi=1, p=1.5)`

ppoisgam(q, mu, phi=1, p=1.5)`rpoisgam(n, mu, phi=1, p=1.5)`

**REQUIRED ARGUMENTS**`x`vector of quantiles. Missing values (NAs) are allowed. `q`vector of quantiles. Missing values (NAs) are allowed. `n`sample size. If length(n) is larger than 1, then length(n) random values are returned. `mu`vector of (positive) means. This is replicated to be the same length as x or q. **OPTIONAL ARGUMENTS**`phi`vector of (positive) dispersion parameters. This is replicated to be the same length as x or q. `p`power index of variance function. Must satisfy 1 <= p <= 2. The variance of the distribution is phi*mu ^{p }**VALUE**- Vector of length n or the same length as
`x`or`q`giving the density (`dpoisgam`), probability (`ppoisgam`) or random sample (`rpoisgam`) for the Poisson gamma distribution with mean`mu`, dispersion`phi`and index`p`. Elements of`x`or`q`that are missing will cause the corresponding elements of the result to be missing. **BACKGROUND**- The Poisson-gamma distribution is also called compound Poisson. It can be represented as
the distribution of
*Y = X*_{1}+ ... + X_{N}where

*X*to_{1}*X*are independent gamma random variables and_{N}*N*is Poisson. Note that the*Y*has mass at zero, but otherwise has a continuous positive distribution. The distribution can be equivalently represented as a Poisson mixture of gamma distributions.The distribution approaches gamma as p -> 2 and phi * Poisson(mu) as p -> 1. The Poison-gamma distribution is a Tweedie distribution with index p between 1 and 2. Since p = 1 corresponds to Poison and p = 2 corresponds to gamma, the Poison-gamma distribution is genuinely intermediate between the Poisson and gamma distributions.

**REFERENCES**- 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. [PDF] - Jørgensen, B. (1987). Exponential dispersion models.
*J. R. Statist. Soc. B*,**49**, 127-162. **SEE ALSO**- Tweedie Distributions, Tweedie family
- The following function is included in the
`poisgam`software distribution, but is not intended to be called directly by users:`ppoiscc`.

**EXAMPLES**`y <- c(-1,0,4,5,10000)``d <- dpoisgam(y,mu=4,phi=1,p=1.6)``p <- ppoisgam(y,mu=4,phi=1,p=1.6)`

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