Skip to main content
SpringerLink
Log in

One-sample tests for a generalized Fréchet variance of a fuzzy random variable

  • Published:
Metrika Aims and scope Submit manuscript

Abstract

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. 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Aumann RJ (1965) Integrals of set-valued functions. J Math Anal Appl 12: 1–12

    Article  MATH  MathSciNet  Google Scholar 

  • Bertoluzza C, Corral N, Salas A (1995) On a new class of distances between fuzzy numbers. Mathw Soft Comput 2: 71–84

    MATH  MathSciNet  Google Scholar 

  • Bickel PJ, Freedman DA (1981) Some asymptotic theory for the bootstrap. Ann Stat 9(6): 1196–1217

    Article  MATH  MathSciNet  Google Scholar 

  • Billingsley P (1986) Probability and measure. Wiley, Chicago

    MATH  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • 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

    MATH  Google Scholar 

  • Fréchet M (1948) Les éléments aléatoires de nature quelconque dan un espace distancié. Ann Inst Henri Poincaré 10: 215–310

    Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Hoeffding W (1948) A class of statistics with asymptotically normal distribution. Ann Math Stat 19: 293–325

    Article  MATH  MathSciNet  Google Scholar 

  • Körner R (2000) An asymptotic α-test for the expectation of random fuzzy variables. J Stat Plan Inference 83: 331–346

    Article  MATH  Google Scholar 

  • Kratschmer V (2001) A unified approach to fuzzy random variables. Fuzzy Sets Syst 123: 1–9

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • Näther W (2000) On random fuzzy variables of second order and their application to linear statistical inference with fuzzy data. Metrika 51: 201–221

    Article  MATH  MathSciNet  Google Scholar 

  • Näther W, Wünche A (2007) On the conditional variance of fuzzy random variables. Metrika 65: 109–122

    Article  MATH  MathSciNet  Google Scholar 

  • Puri M, Ralescu DA (1985) The concept of normality for fuzzy random variables. Ann Probab 13: 1373–1379

    Article  MATH  MathSciNet  Google Scholar 

  • Puri ML, Ralescu DA (1986) Fuzzy random variables. J Math Anal Appl 114: 409–422

    Article  MATH  MathSciNet  Google Scholar 

  • 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

Download references

Author information

Authors and Affiliations

  1. European Centre for Soft Computing, C/Gonzalo Gutiérrez Quirós s/n, 33600, Mieres-Asturias, Spain

    Ana Belén Ramos-Guajardo & Gil González-Rodríguez

  2. Dpto. de Estadística, I.O y D.M, Universidad de Oviedo, C/Calvo Sotelo s/n, 33007, Oviedo, Spain

    Ana Colubi & María Ángeles Gil

Authors
  1. Ana Belén Ramos-Guajardo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ana Colubi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gil González-Rodríguez
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. María Ángeles Gil
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ana Belén Ramos-Guajardo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ramos-Guajardo, A.B., Colubi, A., González-Rodríguez, G. et al. One-sample tests for a generalized Fréchet variance of a fuzzy random variable. Metrika 71, 185–202 (2010). https://doi.org/10.1007/s00184-008-0225-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00184-008-0225-0

Keywords

Search

Navigation