Invariance

Abstract

Many statistical problems exhibit symmetries, which provide natural restrictions to impose on the statistical procedures that are to be employed. Suppose, for example, that $X_1,\ldots ,X_n$ are independently distributed with probability densities $p_{\theta _1}(x_1),\ldots ,p_{\theta _n}(x_n)$. For testing the hypothesis $H:\theta _1=\cdots =\theta _n$ against the alternative that the $\theta$’s are not all equal, the test should be symmetric in $x_1,\ldots ,x_n$, since otherwise the acceptance or rejection of the hypothesis would depend on the (presumably quite irrelevant) numbering of these variables.