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?


So I think that actually the answer is pretty trivial.

If $X_i \sim N(\mu, 1)$ then we can write

$X_i = Y_i + \mu$ where $Y_i \sim N(0,1)$.


$$X_{(n)} - X_{(i)} =_d Y_{(n)} + \mu - Y_{(i)} - \mu =_d Y_{(n)} -Y_{(i)} \perp \mu$$

So the result is trivial... Thought I would post an answer nonetheless as I couldn't find a good one when I searched.


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