A new class of two-sample test statistics is defined. Sufficient conditions for its asymptotic normality under a broad class of alternatives are obtained. Asymptotic efficiencies relative to the parametric competitors for shift in location and change of scale are also obtained. These results are extended to the situations where the sample sizes are random and the observations come in pairs and each pair has the same but unknown bivariate distribution. As a particular case, the asymptotic normality of the Spearmans rank correlation coefficient is established in Section 5 under all alternatives. In a straight forward manner the two-sample results are extended to the c-sample case and the asymptotic efficiency of the test criterion, in order to test for the equality of the underlying distributions is evaluated when the alternatives involve shifts in location parameters or changes in scale parameters.