Two-sample Kolmogorov–Smirnov fuzzy test for fuzzy random variables
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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\alpha $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">α</mi> </math> </EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>p</mi> </math> </EquationSource> </InlineEquation> value is introduced to evaluate the imprecise hypotheses of interest. In this regard, applying an index called credibility degree, the obtained fuzzy <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>p</mi> </math> </EquationSource> </InlineEquation> 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
