Schur-convexity of 2nd order, certain subclass of multivariate arrangement increasing functions with applications in statistics

It is shown that starting with a certain meaningful problem of the type “ranking of populations”, a need arises to employ functions which we call “Schur-convex of 2nd order with respect to two variables”. These functions L(v1,v2,v3,…,vn) are symmetric, and they are characterized in essence by the relation Lv12″−2Lv1v2″+Lv22″≥0. It is shown that this subclass of Schur-convex functions is closely related to a certain subclass of multivariate arrangement increasing functions introduced by Boland and Proschan [P.J. Boland, F. Proschan, Multivariate arrangement increasing functions with applications in probability and statistics, J. Multivariate Anal. (1988) 25 286–298]. This relation allows us to solve a series of statistical problems concerning maximization of the goal function with respect to the risk criterion on the set of permutations of the function’s arguments.