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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
J�rgensen, 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]
 
J�rgensen, 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