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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

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)

 

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