Advertisement

Metrika

, Volume 75, Issue 1, pp 5–22 | Cite as

Fuzzy density estimation

  • Mohsen ArefiEmail author
  • Reinhard Viertl
  • S. Mahmoud Taheri
Article

Abstract

A new approach to density estimation with fuzzy random variables (FRV) is developed. In this approach, three methods (histogram, empirical c.d.f., and kernel methods) are extended for density estimation based on α-cuts of FRVs.

Keywords

Fuzzy density estimation Fuzzy random variable (FRV) Empirical cumulative distribution function (c.d.f.) Histogram method Kernel method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alberts T, Karunamuni RJ (2003) A semiparametric method of boundary correction for kernel density estimation. Stat Probab Lett 61: 287–298CrossRefzbMATHMathSciNetGoogle Scholar
  2. Billingsley P (1995) Probability and measure, 3rd edn. Wiley, New YorkzbMATHGoogle Scholar
  3. Campos VSM, Dorea CCY (2001) Kernel density estimation: the general case. Stat Probab Lett 55: 173–180CrossRefzbMATHMathSciNetGoogle Scholar
  4. Cheng K, Chu C (2004) Semiparametric density estimation under a two-sample density ratio model. Bernoulli 10(4): 583–604CrossRefzbMATHMathSciNetGoogle Scholar
  5. Cheng PE (1994) Nonparametric estimation of mean functionals with data missing at random. J Am Stat Assoc 89: 81–87CrossRefzbMATHGoogle Scholar
  6. Cheng PE, Chu CK (1996) Kernel estimation of distribution functions and quantiles with missing data. Stat Sinica 6: 63–78zbMATHMathSciNetGoogle Scholar
  7. Devroye L, Györfi L (1985) Nonparametric density estimation. Wiley, New YorkzbMATHGoogle Scholar
  8. Fokianos K (2004) Merging information for semiparametric density estimation. J R Stat Soc Series B (Stat Methodol) 66(4): 941–958CrossRefzbMATHMathSciNetGoogle Scholar
  9. Gil MA (2004) Fuzzy random variables: development and state of the art. In: Klement EP, Pap E (eds) Mathematics of fuzzy systems, linz seminar on fuzzy set theory. Linz, Austria, pp 11–15Google Scholar
  10. Hazelton ML (2000) Marginal density estimation from incomplete bivariate data. Stat Probab Lett 47: 75–84CrossRefzbMATHMathSciNetGoogle Scholar
  11. Jones MC (1991) Kernel density estimation for length biased data. Biometrika 78: 511–519CrossRefzbMATHMathSciNetGoogle Scholar
  12. Ker AP, Ergün AT (2005) Empirical Bayes nonparametric kernel density estimation. Stat Probab Lett 75: 315–324CrossRefzbMATHGoogle Scholar
  13. Keziou A, Leoni-Aubin S (2005) Test of homogeneity in semiparametric two-sample density ratio models. C R Acad Sci Paris Ser I Math 340(12): 905–910zbMATHMathSciNetGoogle Scholar
  14. Keziou A, Leoni-Aubin S (2007) On empirical likelihood for semiparametric two-sample density ratio models. J Stat Plan Inference 138: 915–928CrossRefMathSciNetGoogle Scholar
  15. Klement EP, Puri LM, Ralescu DA (1986) Limit theorems for fuzzy random variables. Proc R Soc Lond 407: 171–182CrossRefzbMATHMathSciNetGoogle Scholar
  16. Klir GJ, Yuan B (1995) Fuzzy sets and fuzzy logic, theory and applications. Prentic-Hall, Englewood Cliffs, NJzbMATHGoogle Scholar
  17. Kruse R, Meyer KD (1987) Statistics with vague data. Reidel Publishing Company, Dordrecht, NetherlandsCrossRefzbMATHGoogle Scholar
  18. Lee YK, Choi H, Park BU, Yu KS (2004) Local likelihood density estimation on random fields. Stat Probab Lett 68: 347–357CrossRefzbMATHMathSciNetGoogle Scholar
  19. Li S, Ogura Y (2006) Strong laws of large numbers for independent fuzzy set-valued random variables. Fuzzy Sets Syst 157: 2569–2578CrossRefzbMATHMathSciNetGoogle Scholar
  20. Loquin K, Strauss O (2008) Histogram density estimators based upon a fuzzy partition. Stat Probab Lett 78: 1863–1868CrossRefzbMATHMathSciNetGoogle Scholar
  21. Owen AB (2001) Empirical likelihood. Chapman & Hall/CRC, LondonCrossRefzbMATHGoogle Scholar
  22. Parzen E (1962) On estimation of a probability density function and mode. Ann Math Stat 33: 1065–1076CrossRefzbMATHMathSciNetGoogle Scholar
  23. Prasaka Rao BLS (1983) Nonparametric functional estimation. Academic Press, New YorkGoogle Scholar
  24. Puri ML, Ralescu DA (1986) Fuzzy random variables. J Math Anal Appl 114: 409–422CrossRefzbMATHMathSciNetGoogle Scholar
  25. Qin J (1998) Inferences for case-control and semiparametric two-sample density ratio models. Biometrika 85(3): 619–630CrossRefzbMATHMathSciNetGoogle Scholar
  26. Qin J, Zhang B (2005) Density estimation under a two-sample semiparametric model. Nonparametric Stat 17(6): 665–683CrossRefzbMATHMathSciNetGoogle Scholar
  27. Rosenblatt M (1956) Remarks on some nonparametric estimates of a density function. Ann Math Stat 27: 642–669CrossRefGoogle Scholar
  28. Rosenblatt M (1971) Curve estimates. Ann Math Stat 42: 1815–1842CrossRefzbMATHMathSciNetGoogle Scholar
  29. Shao J (2003) Mathematical statistics, 2nd edn. Springer-Verlag, New YorkCrossRefzbMATHGoogle Scholar
  30. Sheather SJ (2004) Density estimation. Stat Sci 19: 588–597CrossRefzbMATHMathSciNetGoogle Scholar
  31. Silverman BW (1986) Density estimation for statistics and data analysis. Chapman & Hall, New YorkzbMATHGoogle Scholar
  32. Simonoff J (1996) Smoothing methods in statistics. Springer, New YorkCrossRefzbMATHGoogle Scholar
  33. Taheri SM (2003) Trends in fuzzy statistics. Austrian J Stat 32: 239–257Google Scholar
  34. Trutschnig W (2008) A strong consistency result for fuzzy relative frequencies interpreted as estimator for the fuzzy-valued probability. Fuzzy Sets Syst 159: 259–269CrossRefzbMATHMathSciNetGoogle Scholar
  35. Viertl R (1996) Statistical methods for non-precise data. CRC Press, Boca RatonGoogle Scholar
  36. Viertl R (2006) Univariate statistical analysis with fuzzy data. Comput Stat Data Anal 51: 133–147CrossRefzbMATHMathSciNetGoogle Scholar
  37. Viertl R, Hareter D (2006) Beschreibung und Analyse unscharfer Information: Statistische Methoden für unscharfe Daten. Springer, WienzbMATHGoogle Scholar
  38. Wand MP, Jones MC (1995) Kernel smoothing. Chapman & Hall, LondonzbMATHGoogle Scholar
  39. Wu TJ, Chen ChF, Chen HY (2007) A variable bandwidth selector in multivariate kernel density estimation. Stat Probab Lett 77: 462–467CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Mohsen Arefi
    • 1
    Email author
  • Reinhard Viertl
    • 2
  • S. Mahmoud Taheri
    • 1
  1. 1.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran
  2. 2.Technische Universität Wien, Institut für Statistik und WahrscheinlichkeitstheorieWienAustria

Personalised recommendations