In the course, still introducing some concept of dependent distributions, we will talk about the Dirichlet distribution (which is a distribution over the simplex of ). Let denote the Gamma distribution with density (on )

Let denote independent

random variables, with . Then where

has a Dirichlet distribution with parameter . Note that has a distribution in the simplex of ,

and has density

We will write .

The density for different values of can be visualized below, e.g. , with some kind of symmetry,

or and , below

and finally, below,

Note that marginal distributions are also Dirichlet, in the sense that

if

then

if

, and if

, then

‘s have Beta distributions,

See

Devroye (1986) section XI.4, or

Frigyik, Kapila & Gupta (2010) .This distribution might also be called multivariate Beta distribution. In R, this function can be used as follows

> library(MCMCpack)
> alpha=c(2,2,5)
> x=seq(0,1,by=.05)
> vx=rep(x,length(x))
> vy=rep(x,each=length(x))
> vz=1-x-vy
> V=cbind(vx,vy,vz)
> D=ddirichlet(V, alpha)
> persp(x,x,matrix(D,length(x),length(x))

(to plot the density, as figures above). Note that we will come back on that distribution later on so-called Liouville copulas (see also Gupta & Richards (1986)).

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**Tags:** beta, Dirichlet, gamma, Liouville, MAT8886, MAT8886 copulas and extremes, R-english, simplex