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For a vector $X = (x_1, \dots, x_m)$, let $\mathcal{C}(X) = \frac{1}{x_1+\dots+x_m}(x_1, \dots, x_m)$.

If $(X_a, X_b)$ follows a Dirichlet distribution with parameters $(\alpha_{a_1}, \dots, \alpha_{a_k}, \alpha_{b_a}, \dots, \alpha_{b_l})$, $\alpha_{\cdot}>0$. Anybody knows what is the distribution of $\mathcal{C}(X_a)$? Is it also a Dirichlet?

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  • $\begingroup$ Since the sum of $X_a$ is random, it cannot be a Dirichlet. Unless you complement it with $Z=1-\sum_i X_{ai}$. $\endgroup$
    – Xi'an
    May 24, 2016 at 12:04
  • $\begingroup$ Indeed, I've tried to reformulate my question. I think that now makes sense. $\endgroup$
    – marc1s
    May 24, 2016 at 12:18

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A useful representation of a Dirichlet variable $$X=(X_1,\ldots,X_k)\sim\mathcal{D}_k(a_1,\ldots,a_k)$$ [I mean useful for this question) is that it has the same distribution as $$(Y_1,\ldots,Y_k)\big/\sum_{i=1}^k Y_i$$when$$Y_1\sim \mathcal{G}(a_1,1)\,,\ldots\,, Y_k\sim \mathcal{G}(a_k,1)$$are independent Gamma variates.

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  • $\begingroup$ Thank you! At the end, I've found that the distribution of $\mathcal{C}(X_a)$ is also a Dirichlet distribution with parameters $(\alpha_{a_1}, \dots, \alpha_{a_k})$. The property is available in the book of John Aitchison, Statistical Analysis of Compositional Data, Property 3.4 page 60. $\endgroup$
    – marc1s
    May 25, 2016 at 7:49

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