Assume that we have two populations (X1,Y1) and (X2,Y2) satisfying two general nonparametric regression models Yj=mj(Xj)+εj, j=1,2, where m(⋅) is a smooth location function, εj has zero location and the response Yj is possibly right-censored. In this paper, we propose to test the null hypothesis H0:m1=m2 versus the one-sided alternative H1:m1<m2. We introduce two test statistics for which we obtain the asymptotic normality under the null and the alternative hypotheses. Although the tests are based on nonparametric techniques, they can detect any local alternative converging to the null hypothesis at the parametric rate n−1/2. The practical performance of a bootstrap version of the tests is investigated in a simulation study. An application to a data set about unemployment duration times is also included.