Nonparametric statistics, especially the Mann–Whitney–Wilcoxon (MWW) statistic, have gained widespread acceptance, but are by no means the preferred method for statistical analysis in all situations. A main obstacle to their even wider applicability is that the price paid for their distribution-free property is the loss of efficacy. In fact, the MWW statistic, which is an estimate of the functional ∫ −∞ ∞ F(x) dG(x) , has an efficacy which varies with the underlying distributions F( x) and G( x). To improve the efficacy, this dissertation generalizes the classical MWW statistic to estimates of the functional ∫ −∞ ∞ u{F(x)} dv{G(x)} , where the functions u( x) and v( x) are strictly increasing on [0,1]. Statistical properties of this generalization such as asymptotic normality and admissibility are fully investigated. The optimal choices of functions u( x) and v( x) are studied via the tail binomial polynomials and the Pitman asymptotic efficacy criterion. In the one-sample problem, a similar generalization, based upon the functional ∫ −∞ ∞ u(1−F(−x)) dv(F(x)) , extends the Wilcoxon signed rank statistic.