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The Kumaraswamy distribution has cdf $F(x;a,b) = 1-(1-x^a)^b$. Does anyone know any formulas or properties relating to iterations of this on itself, meaning $$ F_i(x;a,b) = 1-(1-F_{i-1}^a)^b$$

If something similar exists for the Beta distribution, I would be interested in that as well.

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  • $\begingroup$ Could you add some information about what you want specifically about the iterations? I've run into these before and could answer you better if I knew what you were looking for. $\endgroup$
    – BSteinhurst
    Commented Oct 11, 2011 at 18:22
  • $\begingroup$ @BSteinhurst: any information at all would be great. But specifically, I am interested in calculating/comparing the mean and variance of the distributions at successive iterations. Ideally, I would love to be able to derive $a(i,a,b) = \mathbb{E}X_i$, $X_i\sim F_i$ but more realistically looking for formal proofs that the mean/variance of $F$ decreases/increases with $i$ for constraints on $a$ and $b$. $\endgroup$
    – OctaviaQ
    Commented Oct 11, 2011 at 19:29
  • $\begingroup$ note: I probably shouldn't have used $a(i,a,b)$ as $a()$ is unrelated to $a$. Pretend I wrote $g(i,a,b)$ or some other unrelated letter. $\endgroup$
    – OctaviaQ
    Commented Oct 11, 2011 at 19:33
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    $\begingroup$ Ahh, I think then what I did may be tangential at best. A student and I came across not these distributions but the equations $x = 1-(1-x^{a})^{b}$ in a percolation problem. So we have some experience with the fixed points of the iteration on numbers. That all can be nicely done with a few pages of calculus. $\endgroup$
    – BSteinhurst
    Commented Oct 11, 2011 at 21:41
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    $\begingroup$ Only for the dependence of the stable point upon a and b, not the location of the stable point itself. I've not added an answer to this question because what I know doesn't really answer the question you asked. If you are interested in this side line feel free to contact me by e-mail though. $\endgroup$
    – BSteinhurst
    Commented Oct 13, 2011 at 3:34

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