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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*mup

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 = X1 + ... + XN

where X1 to XN are independent gamma random variables and 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
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]

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