When are the Mean and the Studentized Differences Independent?

Abstract

Let $X_1, \cdots, X_n$ be i.i.d. rv's. Let further $\bar{X} = \sum X_i/n, S^2 = \sum(X_i - \bar{X})^2$, and $U = ((X_1 - \bar{X})/S, \cdots, (X_n - \bar{X})/S)$. If the variables $X_i$ are normally distributed or distributed as linearly transformed Gamma variables, $\bar{X}$ and $U$ are independent. In this paper we show that also the converse must hold.