dpoig: Compound Poisson-gamma distribution

Description

Density, distribution function and quantile function for for the Poisson-gamma compound probability distribution with parameters frac, rate and rate.

Usage

dpoig(x, frac, rate, shape, log=FALSE)
ppoig(q, frac, rate, shape, lower.tail=TRUE, log.p=FALSE)
qpoig(p, frac, rate, shape, S=30, lower.tail=TRUE, log.p=FALSE)

Arguments

vector of (non-negative integer) quantiles. In the context of species abundance distributions, this is a vector of abundances of species in a sample.

vector of (non-negative integer) quantiles. In the context of species abundance distributions, a vector of abundances of species in a sample.

vector of probabilities.

frac

single numeric '0

rate

vector of (non-negative) rates of the gamma distribution of the sampled population (see details). Must be strictly positive.

shape

the shape parameter of the gamma distribution of the sampled population (see details). Must be positive.

log, log.p

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

lower.tail

logical; if TRUE (default), probabilities are P[X <= x],="" otherwise,="" p[x=""> x].

positive integer; initial value for bisection method. (see note)

Value

(log) density of the (zero-truncated) density

Details

A compound Poisson-gamma distribution is a Poisson probability distribution where its single parameter, the process mean rate, is frac*n, at which n is a random variable with gamma distribution. The density function is given by Green & Plotkin (2007).

In ecology, this distribution gives the probability that a species has an abundance of x individuals in a random sample of a fraction frac of the community. In the community the species abundances are independent random variables that follow a gamma density function.

Hence, a Poisson-gamma distribution is a model for species abundances distributions (SAD) under the assumptions: (a) species abundances in the community are independent identically distributed gamma variables, (b) sampling is a Poisson process with expected value frac*n, (c) the sampling is done with replacement, or the fraction sampled is small enough to approximate a sample with replacement.

The Poisson-gamma distribution is also known as the Negative Binomial distribution. The function dpoig is provided to express the Negative Binomial explicitly as a compound distribution. The Fisher log-series (Fisher 1943) is a limiting case where the dispersion parameter of the Negative Binomial tends to zero. As in the case of the Poisson-exponential, the user should not fit to the Poisson-gamma directly, and should use instead the fitls function.

References

Fisher, R.A, Corbert, A.S. and Williams, C.B. (1943) The Relation between the number of species and the number of individuals in a random sample of an animal population. The Journal of Animal Ecology, 12:42--58.