## One-sample tests for a generalized Fréchet variance of a fuzzy random variable.(English)Zbl 1182.62103

Summary: A procedure to test hypotheses about the population variance of a fuzzy random variable is analyzed. The procedure is based on the theory of UH-statistics [see W. Hoeffding, Ann. Math. Stat. 19, 293–325 (1948; Zbl 0032.04101)]. The variance is defined in terms of a general metric to quantify the variability of the fuzzy values about its (fuzzy) mean. An asymptotic one-sample test in a wide setting is developed and a bootstrap test, which is more suitable for small and moderate sample sizes, is also studied. Moreover, the power function of the asymptotic procedure through local alternatives is analyzed. Some simulations showing the empirical behavior and consistency of both tests are carried out. Finally, some illustrative examples of the practical application of the proposed tests are presented.

### MSC:

 62G10 Nonparametric hypothesis testing 62G20 Asymptotic properties of nonparametric inference 62G09 Nonparametric statistical resampling methods 62G86 Nonparametric inference and fuzziness 65C60 Computational problems in statistics (MSC2010)

Zbl 0032.04101
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### References:

  Aumann RJ (1965) Integrals of set-valued functions. J Math Anal Appl 12: 1–12 · Zbl 0163.06301  Bertoluzza C, Corral N, Salas A (1995) On a new class of distances between fuzzy numbers. Mathw Soft Comput 2: 71–84 · Zbl 0887.04003  Bickel PJ, Freedman DA (1981) Some asymptotic theory for the bootstrap. Ann Stat 9(6): 1196–1217 · Zbl 0472.62054  Billingsley P (1986) Probability and measure. Wiley, Chicago · Zbl 0586.28001  Borovskikh Yu V (1986) Theory of U-statistics in Hilbert spaces. Preprint, No. 86.78. UkrSSR Academy of Sciences, Institute of Mathematics, Kiev  Colubi A, Domínguez-Menchero JS, López-Díaz M, Ralescu R (2002) A D E [0, 1] representation of random upper semicontinuous functions. Proc Am Math Soc 130: 3237–3242 · Zbl 1005.28003  Corbatón JA, Ceballos D (2004) Aplicación del método fuzzy delphi a la predicción bursátil. In: XI congress of international association for fuzzy-set management and economy, University Mediterranea of Reggio Calabria, Italia  Diamond P, Kloeden P (1994) Metric spaces of fuzzy sets. World Scientific, Singapore · Zbl 0873.54019  Fréchet M (1948) Les éléments aléatoires de nature quelconque dan un espace distancié. Ann Inst Henri Poincaré 10: 215–310  Gil MA, Montenegro M, González-Rodríguez G, Colubi A, Casals MR (2006) Bootstrap approach to the multi-sample test of means with imprecise data. Comput Stat Data Anal 51: 148–162 · Zbl 1157.62391  González-Rodríguez G, Colubi A, Trutschnig W (2008) Simulating random upper semicontinuous functions: the case of fuzzy data. Inf Sci. doi: 10.1016/j.ins.2008.10.018  González-Rodríguez G, Montenegro M, Colubi A, Gil MA (2006) Bootstrap techniques and fuzzy random variables: synergy in hypothesis testing with fuzzy data. Fuzzy Sets Syst 157: 2608–2613 · Zbl 1119.62037  Hoeffding W (1948) A class of statistics with asymptotically normal distribution. Ann Math Stat 19: 293–325 · Zbl 0032.04101  Körner R (2000) An asymptotic {$$\alpha$$}-test for the expectation of random fuzzy variables. J Stat Plan Inference 83: 331–346 · Zbl 0976.62013  Kratschmer V (2001) A unified approach to fuzzy random variables. Fuzzy Sets Syst 123: 1–9 · Zbl 1004.60003  Lubiano MA, Alonso C, Gil MA (1999) Statistical inferences on the S-mean squared dispersion of a fuzzy random variable. In: Proceedings of the joint EUROFUSE-SIC99, Budapest, pp 532–537  Montenegro M, Colubi A, Casals MR, Gil MA (2004) Asymptotic and Bootstrap techniques for testing the expected value of a fuzzy random variable. Metrika 59: 31–49 · Zbl 1052.62048  Näther W (2000) On random fuzzy variables of second order and their application to linear statistical inference with fuzzy data. Metrika 51: 201–221 · Zbl 1093.62557  Näther W, Wünche A (2007) On the conditional variance of fuzzy random variables. Metrika 65: 109–122 · Zbl 1105.62017  Puri M, Ralescu DA (1985) The concept of normality for fuzzy random variables. Ann Probab 13: 1373–1379 · Zbl 0583.60011  Puri ML, Ralescu DA (1986) Fuzzy random variables. J Math Anal Appl 114: 409–422 · Zbl 0592.60004  Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning. Part 1. Inf Sci. 8:199–249; part 2. Inf. Sci. 8:301-353; part 3. Inf. Sci. 9:43–80
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