Shape-two Weibull scale likelihood-ratio test (${\sum }_i{X}_i^2\ge ({\lambda }_0^2/2)q_{2n,1-\alpha }$)

For independent Weibull distribution observations of known shape two, the likelihood ratio between any scale ${\lambda }_1>{\lambda }_0$ and the null scale ${\lambda }_0$ increases with ${\sum }_i{X}_i^2$. Under the null, $2{\sum }_i{X}_i^2/{\lambda }_0^2$ has a chi-squared distribution with $2n$ degrees of freedom. The same upper-tail critical region is therefore most powerful for every larger scale, giving a uniformly most powerful size-$\alpha$ test.