Define the Kolmogorov-Smirnov statistic for testing the null hypothesis that real random variables $X_{1}, \ldots, X_{n}$ are independently and identically distributed with specified continuous, strictly increasing distribution function $F$, and show that its null distribution does not depend on $F$.

A composite hypothesis $H_{0}$ specifies that, when the unknown positive parameter $\Theta$ takes value $\theta$, the random variables $X_{1}, \ldots, X_{n}$ arise independently from the uniform distribution $\mathrm{U}[0, \theta]$. Letting $J:=\arg \max _{1 \leqslant i \leqslant n} X_{i}$, show that, under $H_{0}$, the statistic $\left(J, X_{J}\right)$ is sufficient for $\Theta$. Show further that, given $\left\{J=j, X_{j}=\xi\right\}$, the random variables $\left(X_{i}: i \neq j\right)$ are independent and have the $\mathrm{U}[0, \xi]$ distribution. How might you apply the Kolmogorov-Smirnov test to test the hypothesis $H_{0}$ ?