I have $X_1, X_2, ..., X_n $ that are random samples from the single variate $N(\mu,\sigma^2) $. I want to prove that the mean $\bar{X}$ and the sample variance $s_x ^2 = \frac{1}{(n- 1)} \sum_{i=1}^n (X_i - \bar{X}) $ are independent.

As I understand it, I can use Basu's Theorem. Since $\bar{X}$ is a complete sufficient statistic, and $s_x^2$ is an ancillary statistic, Basu's theorem states that they are indeed independent.

I understand why $\bar{X}$ is a complete sufficient statistic logically, but I can't figure out how to show it mathematically. Same goes with the sample variance. Could someone please explain (mathematically) why $\bar{X}$ is a complete sufficient statistic, and $s_x^2$ is an ancillary statistic? In most sources I've checked this fact just seems obvious and they don't show the proof...

  • $\begingroup$ Complete sufficiency of sample mean follows from the fact that the pdf of $X_1,\ldots,X_n$ is a member of exponential family. And sample variance is ancillary because its distribution does not depend on the parameter $\mu$. See math.stackexchange.com/questions/47350/…. $\endgroup$ – StubbornAtom Apr 16 at 14:52

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