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

• Autorzy: Klukowski Leszek

Identyfikatory

DOI

10.2478/candc-2022-0027

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

pairwise comparisons with random errors; estimation of relations; testing of relations; estimation of trees; testing of trees; nearest adjoining order idea

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2022
• Tom: Vol. 51, No. 4
• Strony: 501--522
• Opis fizyczny: Bibliogr. 17 poz., tab.

Twórcy

autor

Klukowski Leszek

• Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland

Bibliografia

• Bradley, R.A. (1976) Science, statistics and paired comparisons. Biometrics, 32, 213–232.
• David, H. A. (1988) The Method of Paired Comparisons, 2nd ed. Ch. Griffin, London.
• Domański, C. (1990) Statistical tests (in Polish). PWE, Warsaw.
• Gordon, A.D. (1999) Classification, 2nd ed. Chapman&Hall/CRC.
• Garfinkel, R. S. and Nemhauser, G. L. (1972) Integer Programming. J Wiley and Son, Inc., New York, London, Sydney, Toronto.
• Hansen, P., Jaumard, B. and Sanlaville, E. (1994) Partitioning Problems in Cluster Analysis: A Review of Mathematical Programming Approaches. Studies in Classification, Data Analysis, and Knowledge Organization. Springer-Verlag.
• Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables. JASA, 58, 13–30.
• Klukowski, L. (2006) Tests for relation type – equivalence or tolerance – in a finite set of elements. Control and Cybernetics 35, 2, 369–84.
• Klukowski, L. (2011) Methods of Estimation of Relations of: Equivalence, Tolerance, and Preference in a Finite Set. IBS PAN, Series: Systems Research, Vol. 69, Warsaw.
• Klukowski, L. (2017) Determining an estimate of an equivalence relation for moderate and large sized sets. Operations Research and Decisions, 27, 2, 45–58.
• Klukowski, L. (2021a) Pairwise comparisons in the form of difference of ranks – testing of estimates of preference relation. Uncertainty and Imprecision in Decision Making and Decision Support – New Challenges, Solutions and Perspectives. Advances in Intelligent Systems and Computing 1081. Springer, 214–224.
• Klukowski, L. (2021b) Statistical tests to verification of estimate of the preference relation resulting from pairwise comparisons. Uncertainty and Imprecision in Decision Making and Decision Support – New Challenges, Solutions and Perspectives Advances in Intelligent Systems and Computing 1081. Springer, 225–238.
• Klukowski, L. (2021c) Estimation and Verification of Relations and Trees on the Basis of Multiple Binary and Multivalent Comparisons. Systems Research Institute, Polish Academy of Sciences, series: Systems Research, Vol. 79, Warsaw.
• Raghavachari, M. (2004) Measures of Concordance for Assessing Agreement in Ratings and Rank Order Data. In: N. Balakrishnan, N. Kannan and H. N. Nagaraja, eds., Advances in Ranking and Selection, Multiple Comparisons, and Reliability. Methodology and Applications. Statistics for Industry and Technology. Birkhäuser, Boston, Basel, Berlin.
• Serfling, R. J. (1980) Approximation Theorems of Mathematical Statistics. J. Wiley and Sons Inc.
• Slater, P. (1961) Inconsistencies in a schedule of paired comparisons. Biometrika, 1961, 48, 303–312.
• Thompson, W. A., Jr. and Remage, R., Jr. (1964) Rankings from paired comparisons. Ann. Math. Statist. 35, 739–47.