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

Mikhail Revyakov

Journal of Multivariate Analysis, 2013, vol. 116, issue C, 25-34

Abstract: 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.

Keywords: Majorization; Ranking of populations; Statistical decision function; Monotone likelihood ratio; Reliability; Tests; Sample mean; Symmetric log-concave density (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1016/j.jmva.2012.11.013

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