Let X1,...,Xm and Y1,...,Yn be two independent samples from continuous distributions F and G respectively. Using a Hoeffding type theorem, the authors obtain the distributions of the vector S=(SÅ(1Å),...,SÅ(nÅ)), where SÅ(jÅ)=ℝ(Xi’s•YÅ(jÅ)) and YÅ(jÅ) is the j-th order statistic of Y sample, under three truncation models: (a) G is a left truncation of F or G is a right truncation of F, (b) F is a right truncation of H and G is a left truncation of H, where H is some continuous distribution function, (c) G is a two tail truncation of F. Exploiting the relation between S and the vector R of the ranks of the order statistics of the Y-sample in the pooled sample, they can obtain exact distributions of many rank tests. The authors use these to compare powers of the Hajek test, the Sidak Vondracek test and the Mann-Whitney-Wilcoxon test. They derive some order relations between the values of the probability-functions under each model. Hence find that the tests based on SÅ(1Å) and SÅ(nÅ) are the UMP rank tests for the alternative (a). The authors also find LMP rank tests under the alternatives (b) and (c).
