# DirMnom: Dirichlet-multinomial (multivariate Polya) distribution

## Description

Density function, cumulative distribution function and random generation for the Dirichlet-multinomial (multivariate Polya) distribution.

## Usage

ddirmnom(x, size, alpha, log = FALSE)rdirmnom(n, size, alpha)

## Arguments

x

$$k$$-column matrix of quantiles.

size

numeric vector; number of trials (zero or more).

alpha

$$k$$-values vector or $$k$$-column matrix; concentration parameter. Must be positive.

log

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

n

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

## Details

If $$(p_1,\dots,p_k) \sim \mathrm{Dirichlet}(\alpha_1,\dots,\alpha_k)$$ and $$(x_1,\dots,x_k) \sim \mathrm{Multinomial}(n, p_1,\dots,p_k)$$, then $$(x_1,\dots,x_k) \sim \mathrm{DirichletMultinomial(n, \alpha_1,\dots,\alpha_k)}$$.

Probability density function $$f(x) = \frac{\left(n!\right)\Gamma\left(\sum \alpha_k\right)}{\Gamma\left(n+\sum \alpha_k\right)}\prod_{k=1}^K\frac{\Gamma(x_{k}+\alpha_{k})}{\left(x_{k}!\right)\Gamma(\alpha_{k})}$$

## References

Gentle, J.E. (2006). Random number generation and Monte Carlo methods. Springer.

Kvam, P. and Day, D. (2001) The multivariate Polya distribution in combat modeling. Naval Research Logistics, 48, 1-17.

Dirichlet, Multinomial