Nonparametric rank-based statistics and significance tests for fuzzy data

  • Authors:

  • Thierry Denœux;Marie-Hélène Masson;Pierre-Alexandre Hébert

  • Affiliations:

  • UMR CNRS-6599 Heudiasyc, Centre de Recherches de Royallieu, Université de Technologie de Compiègne, BP 20529, F-60205 Compiègne Cedex, France;UMR CNRS-6599 Heudiasyc, Centre de Recherches de Royallieu, Université de Technologie de Compiègne, BP 20529, F-60205 Compiègne Cedex, France;UMR CNRS-6599 Heudiasyc, Centre de Recherches de Royallieu, Université de Technologie de Compiègne, BP 20529, F-60205 Compiègne Cedex, France

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2005

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Abstract

Nonparametric rank-based statistics depending only on linear orderings of the observations are extended to fuzzy data. The approach relies on the definition of a fuzzy partial order based on the necessity index of strict dominance between fuzzy numbers, which is shown to contain, in a well-defined sense, all the ordinal information present in the original data. A concept of fuzzy set of linear extensions of a fuzzy partial order is introduced, allowing the approximate computation of fuzzy statistics alpha-cutwise using a Markov Chain Monte Carlo simulation approach. The usual notions underlying significance tests are also extended, leading to the concepts of fuzzy p-value, and graded rejection of the null hypothesis (quantified by a degree of possibility and a degree of necessity) at a given significance level. This general approach is demonstrated in two special cases: Kendall's rank correlation coefficient, and Wilcoxon's two-sample rank sum statistic.