Geodesic
Fuzzy empirical distribution function: Properties and application
Kybernetika, Tome 49 (2013) no. 6, pp. 962-982 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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The concepts of cumulative distribution function and empirical distribution function are investigated for fuzzy random variables. Some limit theorems related to such functions are established. As an application of the obtained results, a method of handling fuzziness upon the usual method of Kolmogorov-Smirnov one-sample test is proposed. We transact the $\alpha$-level set of imprecise observations in order to extend the usual method of Kolmogorov-Smirnov one-sample test. To do this, the concepts of fuzzy Kolmogorov-Smirnov one-sample test statistic and p-value are extended to the fuzzy Kolmogorov-Smirnov one-sample test statistic and fuzzy p-value, respectively. Finally, a preference degree between two fuzzy numbers is employed for comparing the observed fuzzy p-value and the given fuzzy significance level, in order to accept or reject the null hypothesis of interest. Some numerical examples are provided to clarify the discussions in this paper.
The concepts of cumulative distribution function and empirical distribution function are investigated for fuzzy random variables. Some limit theorems related to such functions are established. As an application of the obtained results, a method of handling fuzziness upon the usual method of Kolmogorov-Smirnov one-sample test is proposed. We transact the $\alpha$-level set of imprecise observations in order to extend the usual method of Kolmogorov-Smirnov one-sample test. To do this, the concepts of fuzzy Kolmogorov-Smirnov one-sample test statistic and p-value are extended to the fuzzy Kolmogorov-Smirnov one-sample test statistic and fuzzy p-value, respectively. Finally, a preference degree between two fuzzy numbers is employed for comparing the observed fuzzy p-value and the given fuzzy significance level, in order to accept or reject the null hypothesis of interest. Some numerical examples are provided to clarify the discussions in this paper.
Classification : 62A10, 62G10, 62G86, 93C42, 93C57, 93E12
Keywords: fuzzy cumulative distribution function; fuzzy empirical distribution function; Kolmogorov–Smirnov test; fuzzy p-value; convergence with probability one; degree of accept; degree of reject; Glivenko–Cantelli theorem 
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Hesamian, Gholamreza; Taheri, S. M. Fuzzy empirical distribution function: Properties and application. Kybernetika, Tome 49 (2013) no. 6, pp. 962-982. http://geodesic.mathdoc.fr/item/KYB_2013_49_6_a8/

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