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I am sampling an arcsine distribution, with probability density function

$F(x) = \frac{1}{\pi\sqrt{(x - a)(b-x)}}$

which is defined between $a<x<b$. I want to estimate $a$ and $b$, that is, the minimum and maximum values of the distribution. My approach is to sample the distribution many times and then use the minimum and maximum of the samples as an estimate for the minimum and maximum of the distribution.

My question is: how do I calculate a confidence interval (or other similar metric) for my estimates of the distribution minimum/maximum? There are many examples of how to calculate a confidence interval for the mean, but not for the minimum/maximum.

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    $\begingroup$ It's the same procedure. Generate many samples of size n, save the maximum of each sample, return the 5% to 95% interval (or some other). $\endgroup$
    – user2974951
    Commented May 12, 2022 at 13:12
  • $\begingroup$ You can directly find expressions for the distributions of the maximum and minimum as functions of $a$ and $b.$ The rest is a straightforward application of the definition of a confidence interval. $\endgroup$
    – whuber ♦
    Commented May 12, 2022 at 13:46
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    $\begingroup$ This thing you have stated to be a density doesn't integrate to 1; its normalizing constant should depend on $b-a$. Or if you intended that it not be a normalizzed density, then given it's unnormalized, why the need for $\pi$ in there? $\endgroup$
    – Glen_b
    Commented May 13, 2022 at 3:00

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