# Showing these statistics are ancillary

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?

If $$X_i \sim N(\mu, 1)$$ then we can write
$$X_i = Y_i + \mu$$ where $$Y_i \sim N(0,1)$$.
$$X_{(n)} - X_{(i)} =_d Y_{(n)} + \mu - Y_{(i)} - \mu =_d Y_{(n)} -Y_{(i)} \perp \mu$$