Incomplete U-statistics in two-sample problems

Abstract

Incomplete U-statistics are obtained by sampling summands of a U-statistic. In two-sample cases, a problem of how to derive an optimal incomplete U-statistic is more complicated than that in one-sample cases. In this paper, we consider a sampling scheme to derive incomplete U-statistics in two-sample cases and present an optimal one in four cases which are useful in the practical situations. It's found that the optimal procedures ease us of heavy computational burden with little loss of information.