Kybernetika 31 no. 3, 269-291, 1995

Rank tests for scale: Hájek's influence and recent developments

Hermann Witting


In discussion are recent developments on rank tests for two-sample dispersion problems with continuous one-dimensional distribution functions $F_1$ and $F_2$. It turns out that the nonparametric theory strongly depends on whether the dispersion centers $\mu_1$ and $\mu_2$ are known (and equal) or unknown (and equal, or unequal). The linear rank tests are adjusted to the case "$\mu_1 = \mu_2$ unknown". Most of the literature following the book of Hájek-Šidák [12] tries to extend this theory to the case "$\mu_1 \ne \mu_2$ unknown". As is indicated in Section 1, these results are explicit or implicit within the framework of semiparametric models. Therefore, Section 2 of this paper starts with a nonparametric formulation of the dispersion problem as it is done for the location case in Lehmann [17]. This is followed by a discussion of the most important dispersion orderings. Section 3 gives a complete solution of the testing problem for the case "$\mu_1 = \mu_2$ known". In Sections 4 and 5 these results are extended to the cases "$\mu_1 = \mu_2$ unknown" and "$\mu_1 \ne \mu_2$ unknown". Sections 2 - 5 survey a couple of partially still unpublished results that were received over the last 15 years. They reflect the influence which Hájek's ideas continue to have on the development of nonparametric statistics.