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Let $x_1,\dots,x_n$ be i.i.d. draws from $Beta\left(\frac{k}2,\frac{k-p-1}{2}\right)$. How are the minimum and maximum order statistics distributed, respectively?

I would greatly appreciate a reference if possible. In general I am not familiar with deriving order statistics.

Edit: Given that the beta distribution can be interpreted as a $k$-statistic for the uniform distribution, my guess is that the minimum or maximum of the beta-distribution is distributed according to another beta distribution.

Edit_2: I added a slightly more precise setting that I happen to care about. In the end I am looking for tailbounds for the minimum and the maximum, so whatever form leads to these I will be content with. I am also ultimately interested in the asymptotic case, but it is my next concern, so to speak.

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  • $\begingroup$ You might want to mention what form of an answer you need. In general, the PDF will be a polynomial times a power of an incomplete Beta function, which is likely not to simplify or even be computable except numerically. $\endgroup$ – whuber♦ Jun 12 '15 at 19:38
  • $\begingroup$ @whuber, I tried to answer your question in my second edit. I first of all changed the setting to be a little more specific to my interests. Second of all I am primarily concerned with the tailbound on both of these. Whatever for leads to that will make me happy. Is my argument correct about why it should be a Beta distribution (at least in some cases)? $\endgroup$ – Lepidopterist Jun 12 '15 at 19:47
  • $\begingroup$ I believe your argument applies when (a) $k/2=1$ and you seek the minimum and (b) $(k-p-1)/2=1$ and you seek the maximum. Otherwise it does not apply because the extreme values of a set of middle-order statistics are not the same things as order statistics. $\endgroup$ – whuber♦ Jun 12 '15 at 19:49
  • $\begingroup$ I am interested in the more general case $(k-p-1)/2$ isn't necessarily $1$. In fact, ultimately I want to calculate what happens when $k$ and $p$ go to infinity. $\endgroup$ – Lepidopterist Jun 12 '15 at 19:57
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    $\begingroup$ you should also specify something about the relationship between $k$ and $p$ as they both go to infinity--does one go faster than the other? $\endgroup$ – MichaelChirico Jun 12 '15 at 23:01
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Let the parent random variable $X \sim Beta(a,b)$ with pdf $f(x)$:

http://www.tri.org.au/se/betadistpdf.png

Then, given a sample of size $n$, the pdf of the $1^{st}$ order statistic (sample minimum) is:

http://www.tri.org.au/se/pdfsampleminimumbetadist.png

... and the pdf of the $n^{th}$ order statistic (sample maximum) is:

http://www.tri.org.au/se/pdfsamplemaximumbetadist.png

where:

  • I am using the OrderStat function from the mathStatica package for Mathematica to automate the nitty-gritties

  • Beta[x,a,b] denotes the incomplete Beta function $B_x(a,b) = \int _0^x t^{a-1} (1-t)^{b-1} d t$

  • and the domain of support is (0,1).

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It's not quite the same sequence of Beta variables you want but this paper may be of interest. It's about stick breaking constructions for the Indian Buffet Process with results on the distributions of order stats for samples drawn iid from $Beta(\frac{\alpha}{K},1)$.

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