MAT8886 the Dirichlet distribution

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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 http://freakonometrics.blog.free.fr/public/perso5/diri11.gif). Let http://freakonometrics.blog.free.fr/public/perso5/diri01.gif denote the Gamma distribution with density (on http://freakonometrics.blog.free.fr/public/perso5/diri03.gif)

http://freakonometrics.blog.free.fr/public/perso5/diri02.gif

Let http://freakonometrics.blog.free.fr/public/perso5/diri04.gif denote independent http://freakonometrics.blog.free.fr/public/perso5/diri05.gif
random variables, with http://freakonometrics.blog.free.fr/public/perso5/diri06.gif. Then http://freakonometrics.blog.free.fr/public/perso5/diri07.gif where

http://freakonometrics.blog.free.fr/public/perso5/diri08.gif

has a Dirichlet distribution with parameter http://freakonometrics.blog.free.fr/public/perso5/diri09.gif. Note that http://freakonometrics.blog.free.fr/public/perso5/diri10.gif has a distribution in the simplex of http://freakonometrics.blog.free.fr/public/perso5/diri11.gif,

http://freakonometrics.blog.free.fr/public/perso5/diri40.gif

and has density

http://freakonometrics.blog.free.fr/public/perso5/diri12.gif

We will write http://freakonometrics.blog.free.fr/public/perso5/diri13.gif.

http://freakonometrics.blog.free.fr/public/perso5/diri28.gif http://freakonometrics.blog.free.fr/public/perso5/diri25.gif

http://freakonometrics.blog.free.fr/public/perso5/diri27.gif http://freakonometrics.blog.free.fr/public/perso5/diri26.gif

The density for different values of http://freakonometrics.blog.free.fr/public/perso5/diri20.gif can be visualized below, e.g. http://freakonometrics.blog.free.fr/public/perso5/diri21.gif, with some kind of symmetry,
http://freakonometrics.blog.free.fr/public/perso5/dirichlet222.gif
or http://freakonometrics.blog.free.fr/public/perso5/diri22.gif and http://freakonometrics.blog.free.fr/public/perso5/diri23.gif, below
http://freakonometrics.blog.free.fr/public/perso5/dirichlet522.gif
and finally, below, http://freakonometrics.blog.free.fr/public/perso5/diri24.gif
http://freakonometrics.blog.free.fr/public/perso5/dirichlet225.gif
Note that marginal distributions are also Dirichlet, in the sense that
if

http://freakonometrics.blog.free.fr/public/perso5/diri13.gif


then

http://freakonometrics.blog.free.fr/public/perso5/diri14.gif


if http://freakonometrics.blog.free.fr/public/perso5/diri15.gif, and if http://freakonometrics.blog.free.fr/public/perso5/diri16.gif, then http://freakonometrics.blog.free.fr/public/perso5/diri17.gif‘s have Beta distributions,

http://freakonometrics.blog.free.fr/public/perso5/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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