extraDistr (version 1.10.0)

GammaPoiss: Gamma-Poisson distribution

Description

Probability mass function and random generation for the gamma-Poisson distribution.

Usage

dgpois(x, shape, rate, scale = 1/rate, log = FALSE)
pgpois(q, shape, rate, scale = 1/rate, lower.tail = TRUE, log.p = FALSE)
rgpois(n, shape, rate, scale = 1/rate)

Arguments

x, q

vector of quantiles.

shape, scale

shape and scale parameters. Must be positive, scale strictly.

rate

an alternative way to specify the scale.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are $P[X\le x]$ otherwise, $P[X>x]$.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Gamma-Poisson distribution arises as a continuous mixture of Poisson distributions, where the mixing distribution of the Poisson rate $\lambda$ is a gamma distribution. When $X\sim \mathrm{Poisson}(\lambda )$ and $\lambda \sim \mathrm{Gamma}(\alpha ,\beta )$, then $X\sim \mathrm{GammaPoisson}(\alpha ,\beta )$.

Probability mass function $$f(x)=\frac{\mathrm{\Gamma }(\alpha +x)}{x!\mathrm{\Gamma }(\alpha )}{\left(\frac{\beta }{1+\beta }\right)}^x{(1-\frac{\beta }{1+\beta })}^{\alpha }$$

Cumulative distribution function is calculated using recursive algorithm that employs the fact that $\mathrm{\Gamma }(x)=(x-1)!$. This enables re-writing probability mass function as

$$f(x)=\frac{(\alpha +x-1)!}{x!\mathrm{\Gamma }(\alpha )}{\left(\frac{\beta }{1+\beta }\right)}^x{(1-\frac{\beta }{1+\beta })}^{\alpha }$$

what makes recursive updating from $x$ to $x+1$ easy using the properties of factorials

$$f(x+1)=\frac{(\alpha +x-1)!(\alpha +x)}{x!(x+1)\mathrm{\Gamma }(\alpha )}{\left(\frac{\beta }{1+\beta }\right)}^x\left(\frac{\beta }{1+\beta }\right){(1-\frac{\beta }{1+\beta })}^{\alpha }$$

and let's us efficiently calculate cumulative distribution function as a sum of probability mass functions

$$F(x)=\sum _{k=0}^{x}f(k)$$

See Also

Gamma, Poisson

Examples

x <- rgpois(1e5, 7, 0.002)
xx <- seq(0, 12000, by = 1)
hist(x, 100, freq = FALSE)
lines(xx, dgpois(xx, 7, 0.002), col = "red")
hist(pgpois(x, 7, 0.002))
xx <- seq(0, 12000, by = 0.1)
plot(ecdf(x))
lines(xx, pgpois(xx, 7, 0.002), col = "red", lwd = 2)