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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, 2015 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, 2015 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, 2015 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, 2015 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, 2015 at 23:01

1 Answer 1

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Let the parent random variable $X \sim Beta(a,b)$ with pdf $f(x)$:


(source: tri.org.au)

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


(source: tri.org.au)

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


(source: tri.org.au)

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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