Hi
let x1,x2,...,xn be a random sample from distribution in under function distribution .
F(x)=[(x/θ)]^β , 0≤x<θ
if β is unknown and θ is known find a shortest confidence interval in level (1-α) for β^2

You keep using $ $ like you want to write latex but we actually have latex support here. If you wrap something in [noparse][/noparse] tags then you'll have latex inside those tags. For example [noparse]\(\frac{\alpha}{\beta}\)[/noparse] will generate: \(\frac{\alpha}{\beta}\)

You mean a confidence interval with shortest expected length?

I know that you can use pivotal quantity or inverting a family of tests to construct the confidence interval, and choosing the quantiles optimally to minimize the expected length. However I am not sure if there is a general way to compare for different methods.

Just reread Berger & Casella Ch9 again. Some optimalities, like UMA (by inverting UMP tests) can be achieved but seems they are optimal in a class of intervals only.

BTW if it is your HW question you better show some effort first.