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Two-sample Kolmogorov–Smirnov fuzzy test for fuzzy random variables

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  • Gholamreza Hesamian
  • Jalal Chachi

Abstract

In this paper, a new method is proposed for developing two-sample Kolmogorov–Smirnov test for the case when the data are observations of fuzzy random variables, and the hypotheses are imprecise rather than crisp. In this approach, first a new notion of fuzzy random variables is introduced. Then, the $$\alpha $$ α -pessimistic values of the imprecise observations are transacted to extend the usual method of two-sample Kolmogorov–Smirnov test. To do this, the concepts of fuzzy cumulative distribution function and fuzzy empirical cumulative distribution function are defined. We also develop a well-known large sample property of the classical empirical cumulative distribution function for fuzzy empirical cumulative distribution function. In addition, the Kolmogorov–Smirnov two-sample test statistic is extended for fuzzy random variables. After that, the method of computing the so-called fuzzy $$p$$ p value is introduced to evaluate the imprecise hypotheses of interest. In this regard, applying an index called credibility degree, the obtained fuzzy $$p$$ p value and the crisp significance level are compared. The result provides a fuzzy test function which leads to some degrees to accept or to reject the null hypothesis. Some numerical examples are provided throughout the paper clarifying the discussions made in this paper. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • Gholamreza Hesamian & Jalal Chachi, 2015. "Two-sample Kolmogorov–Smirnov fuzzy test for fuzzy random variables," Statistical Papers, Springer, vol. 56(1), pages 61-82, February.
    • Handle: RePEc:spr:stpapr:v:56:y:2015:i:1:p:61-82
    DOI: 10.1007/s00362-013-0566-2
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    References listed on IDEAS

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    1. Ana Colubi & Renato Coppi & Pierpaolo D’urso & Maria angeles Gil, 2007. "Statistics with fuzzy random variables," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 277-303.
    2. Shapiro, Arnold F., 2009. "Fuzzy random variables," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 307-314, April.
    3. Colubi, Ana & Gonzalez-Rodriguez, Gil, 2007. "Triangular fuzzification of random variables and power of distribution tests: Empirical discussion," Computational Statistics & Data Analysis, Elsevier, vol. 51(9), pages 4742-4750, May.
    4. S. Taheri & G. Hesamian, 2013. "A generalization of the Wilcoxon signed-rank test and its applications," Statistical Papers, Springer, vol. 54(2), pages 457-470, May.
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    Cited by:

    1. Vahid Ranjbar & Gholamreza Hesamian, 2020. "Copula function for fuzzy random variables: applications in measuring association between two fuzzy random variables," Statistical Papers, Springer, vol. 61(1), pages 503-522, February.

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