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2014 | 43 | nr 4 | 577--588
Tytuł artykułu

Estimation of the Partial Order on the Basis of Pairwise Comparisons

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The problem of estimation of the partial order on the basis of multiple pairwise comparisons in binary and multivalent form, with random errors, is investigated. The estimators are based on the idea of the nearest adjoining order (see Slater, 1961; Klukowski 2011). Two approaches are examined: comparisons indicating the direction of preference (binary) and comparisons indicating the difference of ranks (multivalent) - both with possibility of existence of incomparable elements. The properties of estimators and the optimization problems formulated in order to obtain them are similar to those for the case of complete relation. However, the assumptions about the distributions of comparison errors are different - they comprise the case of incomparable elements. (original abstract)
Opis fizyczny
  • Systems Research Institute, Polish Academy of Sciences
  • Bradley, R. A. (1984) Paired comparisons: some basic procedures and examples. In: P. R. Krishnaiah and P. K. Sen, eds., Handbook of Statistics, Vol. 4. Amsterdam: North-Holland, 299-326.
  • David H. A. (1988) The Method of Paired Comparisons, 2nd ed. Ch. Griffin, London.
  • Flinger A. F., Verducci J. S., eds. (1993) Probability Models and Statistical Analyses for Ranking Data. Lecture Notes in Statistics 80, Springer- Verlag, New York, Berlin, Heidelberg.
  • Hansen, P., Jaumard. B., Sanlaville E. (1994) Partitioning Problems in Cluster Analysis: A Review of Mathematical Programming Approaches. In: 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. (1994) Some probabilistic properties of the nearest adjoining order method and its extensions. Annals of Operational Research, 51, 241-261.
  • Klukowski L. (2008) Estimation of the preference relation the basis of multiple pairwise comparisons in the form of differences of ranks. Control and Cybernetics, 37, 711-729.
  • Klukowski, L. (2011) Methods of Estimation of Relations of: Equivalence, Tolerance, and Preference in a Finite Set. IBS PAN, Series: Systems Research, Volume 69, Warsaw.
  • Klukowski L. (2012) Properties of estimators of the preference relation based on pairwise comparisons - simulation survey. In: K. T. Atanassov et al., eds., New Developments in Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets and Related Topics, 75-90. Volume II: Applications, SRI PAS, Warsaw.
  • Slater P. (1961) Inconsistencies in a schedule of paired comparisons. Biometrika, 48, 303-312.
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