Hermann Witting

# Abstract:

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 Sections 2 - 5 survey a couple of partially still unpublished results that were received over the last 15 years.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.

62G10