Extreme Mass Distributions For K-Increasing Quasi-Copulas

Abstract: The rating of quasi-copula problems in the dependence modeling community has recently risen in spite of the lack of probability interpretation of quasi-copulas. The trendsetting paper J.J. Arias-Garcia, R. Mesiar, and B. De Baets, The unwalked path between quasi-copulas and copulas: Stepping stones in higher dimensions, Internat. J. of Approx. Reasoning, 80 (2017) 89--99, proposes the k-increasing property for some k {\le} d as a property of d-variate quasi-copulas that would shed some light on what is in-between. This hierarchy of classes extends the bivariate notion of supermodularity property. The same authors propose a number of open problems in the continuation of this paper (Fuzzy Sets and Systems 393 (2020), 1--28). Their Open problem 5 asks for the extreme values of the mass distributions associated with multivariate quasi-copulas and was recently solved by the authors of this paper (Fuzzy Sets and Systems 527 (2026) 109698). The main goal of the present paper is to solve the maximal-volume problem (in absolute value) within each of the previously mentioned subclasses. By formulating and solving suitably simplified primal and dual linear programs, we derive the exact maximal negative and positive masses together with the corresponding extremal boxes.