I have the following homework question.
Consider the location-scale family generated by a fixed, specified, distribution function $F$ on $\mathbb{R}$, so
$$F_{\mu,\sigma}(x)=F\left(\frac{x-\mu}{\sigma}\right), \mu\in\mathbb{R}, \sigma>0.$$
Let $\hat{\mu}_n=t_n(x_1,\ldots,x_n)$ be a location-scale equivariant estimate for $\mu$, and let $\hat{\sigma}_n=s_n(x_1,\ldots,x_n)$ be a location invariant and scale equivariant estimate for $\sigma$. Consider the composite hypothesis of the type: $F$ belongs to the class $\{F_{\mu,\sigma}: \mu\in\mathbb{R}, \sigma>0\}$. Show that the distribution of
$$K_n=\sup_{x\,\in\,\mathbb{R}}\left|F_{\hat{\mu}_n,\hat{\sigma}_n}(x)-\mathbb{F}_n(x)\right|$$ does not depend on the underlying $\mu$ and $\sigma$.
My attempt so far: I know that $\mathbb{F}_n$ is the empirical distribution. The "invariant" and "equivariant" descriptions of the estimators mean that for all $\alpha\in \mathbb{R}$ and $\beta>0$,
$$t_n(\alpha+\beta x_1,\ldots,\alpha+\beta x_n)=\alpha+\beta t_n(x_1,\ldots,x_n),\\ s_n(\alpha+\beta x_1,\ldots,\alpha+\beta x_n)=\beta s_n(x_1,\ldots,x_n).$$
I think I have to show that if I have a sample from a distribution $F_{\mu_1,\sigma_1}$ and a second sample from a distribution $F_{\mu_2,\sigma_2}$, then I end up with the same distribution for the value of $K_n$, but I don't know how to proceed from there.