An exact rank test for scale under normality using Helmert's transformation (Q1072305)

Let \(X_ 1,...,X_ m\) and \(X_{m+1},...,X_ n\) be two samples from continuous location/scale-type distributions. For testing equality of scale a rank-type test based on transformed rv's \(\hat X_ j\) is suggested instead of the classical F-test. The rank tests based on \(\hat X_ j\) and on ideally centered \(X_ j\) are shown to be asymptotically equivalent. An additional condition ensures that the result remains true for exact rather than approximate scores. The condition is met by the normal distribution.