Depth functions provide measures of the deepness of a point with respect to a given set of observations. This non-parametric concept can be applied in spaces of any dimension and entails a center-outward ordering for the given data.

Liu and Singh (1993) presented a two sample test that is based on depth-ranks and offers opportunities for further investigations: Compared to the results of Zuo and He (2006) for this test, we make some different and new assumptions to fill a gap in their proof. Observing that the corresponding test statistic is not symmetric with respect to the two samples, some first numerical simulations indicate that the power can be greatly increased if Q(F,G) and Q(G,F) are jointly considered. Furthermore, within the last years, depths with respect to functional data have been established.

Our current research focuses on proving the required properties of such depth functions.