Show independence without using Basu theorem

Question

$X_1,...,X_n,i.i.d,X_1\sim N(\mu,\sigma^2)$

prove that $\xi=f(X_1,...,X_n)$ and $\bar X$ are independent, where $f(X_1,...,X_n)=f(X_1+c,...,X_n+c)$ for any constant $c$

I know this can be done by using Basu theorem, and the book says an orthogonal transform will work too, but I can't figure out how.

Answer 1

Note that we have $f(X_1,\dots,X_n) = f(X_1 - \bar{X},\dots,X_n - \bar{X})$, where of course, $\bar{X} = \frac{1}{n}\sum_{i=1}^n X_i .$ So it is sufficient to show that $\bar{X}$ and $X_1 - \bar{X},\dots,X_n - \bar{X}$ are independent.

From joint normality and the fact that $\text{Cov}(\bar{X},X_i - \bar{X}) = 0$ for $1 \leq i \leq n$ it follows that $\bar{X}$ and $\sum_{i=1}^n l_i (X_i - \bar{X})$ are independent (for any choice of $l_1,\dots, l_n$) and hence $\bar{X}$ and $X_1 - \bar{X},\dots,X_n - \bar{X}$ are independent.