the third statistics independent with sample mean and variance

As we know, if X1, X2 ... XN are IID normal distribution, then the mean of the samples and the variance of the samples are independent.
My question is, could we use X1, X2 ... XN to construct the third statistics, which independent with the sample mean and variance.


TS Contributor
I try to construct one.

We know that the normal family is an exponential family, and the natural statistics: sample mean and sample variance are complete, and sufficient for \( \mu \) and \( \sigma^2 \) respectively. By Basu's Theorem any ancillary statistic is independent of the complete sufficient statistic. Since \( \mu \) is a location parameter, and \( \sigma^2 \) is a scale parameter, the statistic

\( \frac {X_1 - X_2} {X_3 - X_4} \sim \text{Cauchy} \) and is ancillary to \( \mu \) and \( \sigma^2 \), which is independent to both sample mean and sample variance.

Of course we are excluding the trivial statistic, e.g. a constant.