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

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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 \(\alpha \)-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 \(p\) value is introduced to evaluate the imprecise hypotheses of interest. In this regard, applying an index called credibility degree, the obtained fuzzy \(p\) 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.

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Acknowledgments

The authors are very grateful to the Editor-in-Chief, Professor Christine H. Müller, and the anonymous referees, for their constructive comments and suggestions that led to an improved version of this paper.

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Authors and Affiliations

  1. Department of Statistics, Payame Noor University, 19395-3697 , Tehran, Iran

    Gholamreza Hesamian

  2. Department of Mathematics, Statistics and Computer Sciences, Semnan University, 35195-363 , Semnan, Iran

    Jalal Chachi

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  1. Gholamreza Hesamian
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  2. Jalal Chachi
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Correspondence to Gholamreza Hesamian.

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Hesamian, G., Chachi, J. Two-sample Kolmogorov–Smirnov fuzzy test for fuzzy random variables. Stat Papers 56, 61–82 (2015). https://doi.org/10.1007/s00362-013-0566-2

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