the Dirichlet distribution

In the course, since we are still introducing some concepts of dependent distributions, we will talk about the Dirichlet distribution, which is a distribution over the simplex of http://freakonometrics.hypotheses.org/files/2017/07/diri11.gif. Let http://freakonometrics.hypotheses.org/files/2017/07/diri01.gif denote the Gamma distribution with density (on http://freakonometrics.hypotheses.org/files/2017/07/diri03.gif)

http://freakonometrics.hypotheses.org/files/2017/07/diri02.gif

Let http://freakonometrics.hypotheses.org/files/2017/07/diri04.gif denote independent http://freakonometrics.hypotheses.org/files/2017/07/diri05.gif random variables, with http://freakonometrics.hypotheses.org/files/2017/07/diri06.gif. Then http://freakonometrics.hypotheses.org/files/2017/07/diri07.gif where

http://freakonometrics.hypotheses.org/files/2017/07/diri08.gif

has a Dirichlet distribution with parameter

http://freakonometrics.hypotheses.org/files/2017/07/diri09.gif

Note that http://freakonometrics.hypotheses.org/files/2017/07/diri10.gif has a distribution in the simplex of http://freakonometrics.hypotheses.org/files/2017/07/diri11.gif,

http://freakonometrics.hypotheses.org/files/2017/07/diri40.gif

and has density

http://freakonometrics.hypotheses.org/files/2017/07/diri12.gif

We will write http://freakonometrics.hypotheses.org/files/2017/07/diri13.gif.

The density for different values of http://freakonometrics.hypotheses.org/files/2017/07/diri20.gif can be visualized below, e.g. http://freakonometrics.hypotheses.org/files/2017/07/diri21.gif, with some kind of symmetry,
http://freakonometrics.hypotheses.org/files/2017/07/dirichlet222.gif
or http://freakonometrics.hypotheses.org/files/2017/07/diri22.gif and http://freakonometrics.hypotheses.org/files/2017/07/diri23.gif, below
http://freakonometrics.hypotheses.org/files/2017/07/dirichlet522.gif
and finally, below, http://freakonometrics.hypotheses.org/files/2017/07/diri24.gif


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

http://freakonometrics.hypotheses.org/files/2017/07/diri13.gif

then

http://freakonometrics.hypotheses.org/files/2017/07/diri14.gif

if http://freakonometrics.hypotheses.org/files/2017/07/diri15.gif, and if http://freakonometrics.hypotheses.org/files/2017/07/diri16.gif, then http://freakonometrics.hypotheses.org/files/2017/07/diri17.gif‘s have Beta distributions,

http://freakonometrics.hypotheses.org/files/2017/07/diri18.gif

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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