Let $Z_i = X_{(n)} - X_{(i)}$ for $i=1,2,\dots,n$ where $X \sim N(\mu, 1)$, and $X_{(i)}$ is the ith order statistic of the sample.
I want to show $Z=(Z_1,\dots,Z_{n-1})$ are ancillary for $\mu$.
My attempt
I know to do this I need to show that the joint distribution of $(Z_1, \dots, Z_{n-1}) $ is independent of $ \mu$. So essentially I only need to derive the joint distribution and see that $\mu$ does not appear anywhere in it.
We know that $(X_{(1)},\dots,X_{(n)}) \sim n! f_X (X_{(1)},\dots,X_{(n)})$ where $f_X$ is the density function of the original unordered sample.
So my thinking was to find the joint distribution of $(Z_1, \dots, Z_{(n-1)},X_{(n)})$ using the Jacobian transformation method, and then integrate out $X_{(n)}$. However, this results in an integral involving the normal PDF, so I'm not sure it has a closed form way of integrating it?
Is there an easier way to show $Z$ is ancillary than explicitly deriving its joint distribution?