Estimators of the relations of: equivalence, tolerance and preference on the basis of pairwise comparisons with random errors

This article presents a review of results of the author (Klukowski 2011a) in the area of estimation of the relations of: equivalence, tolerance and preference within a finite set, on the basis of multiple, independent (in a stochastic way) pairwise comparisons with random errors, in binary and multivalent forms. These estimators require weaker assumptions than those used in the literature on the subject. Estimates of the relations are obtained on the basis of solutions to problems from discrete optimization. They allow application of both types of comparisons – binary and multivalent (this fact relates to the tolerance and preference relations). The estimates can be verified in a statistical way; in particular, it is possible to verify the type of the relation. The estimates have been applied (by the author) in problems regarding: forecasting, financial engineering and bio-cybernetics