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.