let $x_1,x_2, \ldots, x_n$ be a random sample from distribution under function distribution:
$$F(x)= \left( \frac{x}{\theta} \right)^\beta, \quad 0 \leq x \lt \theta.$$
Where $β$ is unknown but $θ$ is known. find a shortest confidence interval in level $(1-\alpha)$ for $\beta^2$.