Linear rank statistics for the two-sample problem are considered. Under general conditions on the score function and the distribution of the observations large deviation asymptotics for these statistics under the alternative are obtained. After using the Sanov principle an external problem of minimization of the Kullback-Leibler information is solved by means of the calculus of variations and nonlinear analysis. As an application Hodges-Lehmann and Chernoff efficiencies are evaluated. It is shown that they coincide locally with the Pitman, Bahadur and intermediate efficiencies.