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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 + ... + X_N

where X₁ to X_N 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

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)

 

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