A backward-induction algorithm for computing the best convex contrast of two bivariate samples

For real-valued x1, x2 , ..., xn with real-valued “responses” y1, y2 , ...,yn and “scores” s1, s2, ..., sn we solve the problem of computing the maximum of C(k) = Σh=1,...,n sh I[yh ≥ k(xh)] over all convex functions k on R1. The article describes a recursive relation and an algorithm based on it to compute this value and an optimal k in O(n3) steps. For a special choice of scores, max C(k) can be interpreted as a generalized (one-sided) Kolmogorov-Smirnov statistic to test for treatment effect in nonparametric analysis of covariance.