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.
Best Answer
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
OrderStatfunction from the mathStatica package for Mathematica to automate the nitty-grittiesBeta[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).