In this paper the problem of assessing the similarity of two cumulative distribution functions F and G is considered. An asymptotic test based on an α-trimmed version of Mallows distance Γα(F, G) between F and G is suggested, thus demonstrating the similarity of F and G within a preassigned Γα(F, G) neighbourhood at a controlled type I error rate. The test proposed is applied to the validation of goodness of fit and for the nonparametric assessment of bioequivalence. It is shown that Γα(F, G) can be interpreted as average and population equivalence. Our approach is illustrated by various examples.