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Estimating and testing relations and trees on the basis of pairwise comparisons

Klukowski Leszek

Control and Cybernetics

2022

Vol. 51, No. 4

501--522

The paper presents an ample account on author’s results, concerning the methods of estimating and testing of relations (equivalence, tolerance, preference) and trees on the basis of multiple pairwise comparisons with random errors, which are presented in a new book, Klukowski (2021c). The book contains also main results presented earlier in Klukowski (2011). The new content pertains mainly to: the preference relation with incomparable elements, trees and tests for verification of estimates of the relations. The estimators and tests have good statistical properties; estimators - consistency with good convergence, tests - known distributions of test statistics (exact, limiting or approximate). Some properties of estimates (precision and convergence) have been determined using the simulation approach. The results here reported gain in significance in view of the renewed interest in practical application of pairwise comparisons in various application domains, which give rise to various concrete data analysis problems.

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

Klukowski L.

Operations Research and Decisions

2012

Vol. 22, No. 2

15--34

This paper presents a review of results of the author in the area of estimation of the relations of equivalence, tolerance and preference within a finite set based on 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 based on 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 to problems regarding forecasting, financial engineering and bio-cybernetics.