Skip to main content
SpringerLink
Log in

Fuzzy density estimation

  • Published:
Metrika Aims and scope Submit manuscript

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.

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

  • Alberts T, Karunamuni RJ (2003) A semiparametric method of boundary correction for kernel density estimation. Stat Probab Lett 61: 287–298

    Article  MATH  MathSciNet  Google Scholar 

  • Billingsley P (1995) Probability and measure, 3rd edn. Wiley, New York

    MATH  Google Scholar 

  • Campos VSM, Dorea CCY (2001) Kernel density estimation: the general case. Stat Probab Lett 55: 173–180

    Article  MATH  MathSciNet  Google Scholar 

  • Cheng K, Chu C (2004) Semiparametric density estimation under a two-sample density ratio model. Bernoulli 10(4): 583–604

    Article  MATH  MathSciNet  Google Scholar 

  • Cheng PE (1994) Nonparametric estimation of mean functionals with data missing at random. J Am Stat Assoc 89: 81–87

    Article  MATH  Google Scholar 

  • Cheng PE, Chu CK (1996) Kernel estimation of distribution functions and quantiles with missing data. Stat Sinica 6: 63–78

    MATH  MathSciNet  Google Scholar 

  • Devroye L, Györfi L (1985) Nonparametric density estimation. Wiley, New York

    MATH  Google Scholar 

  • Fokianos K (2004) Merging information for semiparametric density estimation. J R Stat Soc Series B (Stat Methodol) 66(4): 941–958

    Article  MATH  MathSciNet  Google Scholar 

  • 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 11–15

    Google Scholar 

  • Hazelton ML (2000) Marginal density estimation from incomplete bivariate data. Stat Probab Lett 47: 75–84

    Article  MATH  MathSciNet  Google Scholar 

  • Jones MC (1991) Kernel density estimation for length biased data. Biometrika 78: 511–519

    Article  MATH  MathSciNet  Google Scholar 

  • Ker AP, Ergün AT (2005) Empirical Bayes nonparametric kernel density estimation. Stat Probab Lett 75: 315–324

    Article  MATH  Google Scholar 

  • 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): 905–910

    MATH  MathSciNet  Google Scholar 

  • Keziou A, Leoni-Aubin S (2007) On empirical likelihood for semiparametric two-sample density ratio models. J Stat Plan Inference 138: 915–928

    Article  MathSciNet  Google Scholar 

  • Klement EP, Puri LM, Ralescu DA (1986) Limit theorems for fuzzy random variables. Proc R Soc Lond 407: 171–182

    Article  MATH  MathSciNet  Google Scholar 

  • Klir GJ, Yuan B (1995) Fuzzy sets and fuzzy logic, theory and applications. Prentic-Hall, Englewood Cliffs, NJ

    MATH  Google Scholar 

  • Kruse R, Meyer KD (1987) Statistics with vague data. Reidel Publishing Company, Dordrecht, Netherlands

    Book  MATH  Google Scholar 

  • Lee YK, Choi H, Park BU, Yu KS (2004) Local likelihood density estimation on random fields. Stat Probab Lett 68: 347–357

    Article  MATH  MathSciNet  Google Scholar 

  • Li S, Ogura Y (2006) Strong laws of large numbers for independent fuzzy set-valued random variables. Fuzzy Sets Syst 157: 2569–2578

    Article  MATH  MathSciNet  Google Scholar 

  • Loquin K, Strauss O (2008) Histogram density estimators based upon a fuzzy partition. Stat Probab Lett 78: 1863–1868

    Article  MATH  MathSciNet  Google Scholar 

  • Owen AB (2001) Empirical likelihood. Chapman & Hall/CRC, London

    Book  MATH  Google Scholar 

  • Parzen E (1962) On estimation of a probability density function and mode. Ann Math Stat 33: 1065–1076

    Article  MATH  MathSciNet  Google Scholar 

  • Prasaka Rao BLS (1983) Nonparametric functional estimation. Academic Press, New York

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Qin J (1998) Inferences for case-control and semiparametric two-sample density ratio models. Biometrika 85(3): 619–630

    Article  MATH  MathSciNet  Google Scholar 

  • Qin J, Zhang B (2005) Density estimation under a two-sample semiparametric model. Nonparametric Stat 17(6): 665–683

    Article  MATH  MathSciNet  Google Scholar 

  • Rosenblatt M (1956) Remarks on some nonparametric estimates of a density function. Ann Math Stat 27: 642–669

    Article  Google Scholar 

  • Rosenblatt M (1971) Curve estimates. Ann Math Stat 42: 1815–1842

    Article  MATH  MathSciNet  Google Scholar 

  • Shao J (2003) Mathematical statistics, 2nd edn. Springer-Verlag, New York

    Book  MATH  Google Scholar 

  • Sheather SJ (2004) Density estimation. Stat Sci 19: 588–597

    Article  MATH  MathSciNet  Google Scholar 

  • Silverman BW (1986) Density estimation for statistics and data analysis. Chapman & Hall, New York

    MATH  Google Scholar 

  • Simonoff J (1996) Smoothing methods in statistics. Springer, New York

    Book  MATH  Google Scholar 

  • Taheri SM (2003) Trends in fuzzy statistics. Austrian J Stat 32: 239–257

    Google Scholar 

  • Trutschnig W (2008) A strong consistency result for fuzzy relative frequencies interpreted as estimator for the fuzzy-valued probability. Fuzzy Sets Syst 159: 259–269

    Article  MATH  MathSciNet  Google Scholar 

  • Viertl R (1996) Statistical methods for non-precise data. CRC Press, Boca Raton

    Google Scholar 

  • Viertl R (2006) Univariate statistical analysis with fuzzy data. Comput Stat Data Anal 51: 133–147

    Article  MATH  MathSciNet  Google Scholar 

  • Viertl R, Hareter D (2006) Beschreibung und Analyse unscharfer Information: Statistische Methoden für unscharfe Daten. Springer, Wien

    MATH  Google Scholar 

  • Wand MP, Jones MC (1995) Kernel smoothing. Chapman & Hall, London

    MATH  Google Scholar 

  • Wu TJ, Chen ChF, Chen HY (2007) A variable bandwidth selector in multivariate kernel density estimation. Stat Probab Lett 77: 462–467

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran

    Mohsen Arefi & S. Mahmoud Taheri

  2. Technische Universität Wien, Institut für Statistik und Wahrscheinlichkeitstheorie, Wiedner Hauptstrasse 8-10/107, 1040, Wien, Austria

    Reinhard Viertl

Authors
  1. Mohsen Arefi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Reinhard Viertl
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Mahmoud Taheri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohsen Arefi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arefi, M., Viertl, R. & Taheri, S.M. Fuzzy density estimation. Metrika 75, 5–22 (2012). https://doi.org/10.1007/s00184-010-0311-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00184-010-0311-y

Keywords

Search

Navigation