Abstract Let (x 1, y 1),…, (x n, y n) be a sample of points in , consisting of two subsamples (x 1, y 1),…, (x n1, y n1) and (x n1+1, y n1+1),…, (x n, y n), where n = n 1 + n 2. We consider the problem of separating the two subsamples by a convex contrast curve k…that is, a real-valued convex function x → k(x). For given curve k we consider the relative frequency p 1 of points of the first sample above the graph of k similarly p 2. The goal is to choose k such that the difference in relative frequency p 1 – p 2 becomes maximal. This maximal value can be regarded as a generalized one-sided Kolmogorov-Smirnov test statistic , where p 1(A) = (1/n 1)#{i ≤ n 1 : (xi, y i) ∈ A}, p 2(A) = (1/n 2#{n 1 + 1 ≤ i ≤ n : (x i, y i) ∈ A} and is the class of all epigraphs {(x y) : k(x) ≤ y} of convex functions k. The standard Kolmogorov-Smirnov test statistic corresponds to the class consisting of all sets —that is, epigraphs of constant functions. Test statistics of this form arise in nonparametric analysis of covarianc...