maximum of a Dirichlet vector

An intriguing question on Stack Exchange this weekend, about the distribution of max{p¹,p²,…}the maximum component of a Dirichlet vector Dir(a¹,a²,…) with arbitrary hyper-parameters. Writing the density of this random variable is feasible, using its connection with a Gamma vector, but I could not find a closed-form expression. If there is such an expression, it may follow from the many properties of the Dirichlet distribution and I’d be interested in learning about it. (Very nice stamp, by the way! I wonder if the original formula was made with LaTeX…)

This entry was posted on September 26, 2016 at 2:18 pm and is filed under Books, Statistics with tags cross validated, Dirichlet distribution, LaTeX, marginalisation, order statistics, Peter Dirichlet, Stack Exchange, stamp. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

4 Responses to “maximum of a Dirichlet vector”

Teddy Says:
August 18, 2020 at 3:24 pm

What if we assume a flat Dirichlet? a_1 = a_2 = … = 1

Reply
- xi'an Says:
  August 18, 2020 at 7:02 pm
  
  Sure, for such a special case, there is a resolution.
  
  Reply
Georges Henry Says:
September 26, 2016 at 5:22 pm

Sans espoir. Meme pour une Dirichlet d’ordre 2 tu demandes le calcul explicite de

$\int_c^{1/c}\dfrac{t^{a_1-1}}{(1+t)^{a_1+a_2}}dt$

pas faisable en general.

Reply
- xi'an Says:
  September 26, 2016 at 5:33 pm
  
  Merci de cette prompte réponse!
  
  Reply