IDEAS home Printed from https://ideas.repec.org/a/eee/csdana/v77y2014icp130-145.html
   My bibliography  Save this article

Strong consistency and rates of convergence for a random estimator of a fuzzy set

Author

Listed:
  • Terán, Pedro
  • López-Díaz, Miguel

Abstract

An approximation scheme for estimating a fixed, unknown fuzzy set from random samples taken from the nested random set defined by its α-level sets is presented. Its strong consistency is studied, giving rates of convergence in four metrics. A simulation study suggests that the behaviour for moderately small samples is coherent with the theoretical rate of convergence valid for large samples.

Suggested Citation

  • Terán, Pedro & López-Díaz, Miguel, 2014. "Strong consistency and rates of convergence for a random estimator of a fuzzy set," Computational Statistics & Data Analysis, Elsevier, vol. 77(C), pages 130-145.
    • Handle: RePEc:eee:csdana:v:77:y:2014:i:c:p:130-145
    DOI: 10.1016/j.csda.2014.02.016
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167947314000565
    Download Restriction: Full text for ScienceDirect subscribers only.
    File URL: https://libkey.io/10.1016/j.csda.2014.02.016?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Baíllo, Amparo & Cuesta-Albertos, Juan A. & Cuevas, Antonio, 2001. "Convergence rates in nonparametric estimation of level sets," Statistics & Probability Letters, Elsevier, vol. 53(1), pages 27-35, May.
    2. Hall, Peter & Nussbaum, Michael & Stern, Steven E., 1997. "On the Estimation of a Support Curve of Indeterminate Sharpness," Journal of Multivariate Analysis, Elsevier, vol. 62(2), pages 204-232, August.
    3. Korostelev, A. P. & Simar, L. & Tsybakov, A. B., 1995. "Estimation of monotone boundaries," LIDAM Reprints CORE 1178, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Norberg, Tommy, 1987. "Semicontinuous processes in multi-dimensional extreme value theory," Stochastic Processes and their Applications, Elsevier, vol. 25, pages 27-55.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Abdelaati Daouia & Hohsuk Noh & Byeong U. Park, 2016. "Data envelope fitting with constrained polynomial splines," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 3-30, January.
    2. Leonie Selk & Charles Tillier & Orlando Marigliano, 2022. "Multivariate boundary regression models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(1), pages 400-426, March.
    3. Daouia, Abdelaati & Florens, Jean-Pierre & Simar, Léopold, 2009. "Frontier Estimation and Extreme Values Theory," IDEI Working Papers 611, Institut d'Économie Industrielle (IDEI), Toulouse.
    4. Klemelä, Jussi, 2004. "Complexity penalized support estimation," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 274-297, February.
    5. Cheng, Ming-Yen & Hall, Peter, 2006. "Methods for tracking support boundaries with corners," Journal of Multivariate Analysis, Elsevier, vol. 97(8), pages 1870-1893, September.
    6. Girard, Stéphane & Jacob, Pierre, 2008. "Frontier estimation via kernel regression on high power-transformed data," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 403-420, March.
    7. Abdelaati Daouia & Léopold Simar & Paul W. Wilson, 2017. "Measuring firm performance using nonparametric quantile-type distances," Econometric Reviews, Taylor & Francis Journals, vol. 36(1-3), pages 156-181, March.
    8. Pham, Manh D. & Zelenyuk, Valentin, 2019. "Weak disposability in nonparametric production analysis: A new taxonomy of reference technology sets," European Journal of Operational Research, Elsevier, vol. 274(1), pages 186-198.
    9. McDonald, John, 2009. "Using least squares and tobit in second stage DEA efficiency analyses," European Journal of Operational Research, Elsevier, vol. 197(2), pages 792-798, September.
    10. Léopold Simar & Valentin Zelenyuk, 2007. "Statistical inference for aggregates of Farrell-type efficiencies," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 22(7), pages 1367-1394.
    11. Girard, Stéphane & Guillou, Armelle & Stupfler, Gilles, 2013. "Frontier estimation with kernel regression on high order moments," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 172-189.
    12. Kneip, Alois & Simar, Léopold & Wilson, Paul W., 2008. "Asymptotics And Consistent Bootstraps For Dea Estimators In Nonparametric Frontier Models," Econometric Theory, Cambridge University Press, vol. 24(6), pages 1663-1697, December.
    13. Leopold Simar & Valentin Zelenyuk, 2006. "On Testing Equality of Distributions of Technical Efficiency Scores," Econometric Reviews, Taylor & Francis Journals, vol. 25(4), pages 497-522.
    14. Goldenshluger, A. & Tsybakov, A., 2004. "Estimating the endpoint of a distribution in the presence of additive observation errors," Statistics & Probability Letters, Elsevier, vol. 68(1), pages 39-49, June.
    15. Kneip, Alois & Simar, Léopold & Van Keilegom, Ingrid, 2015. "Frontier estimation in the presence of measurement error with unknown variance," Journal of Econometrics, Elsevier, vol. 184(2), pages 379-393.
    16. Daouia, Abdelaati & Laurent, Thibault & Noh, Hohsuk, 2017. "npbr: A Package for Nonparametric Boundary Regression in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 79(i09).
    17. Dariush Khezrimotlagh & Wade D. Cook & Joe Zhu, 2021. "Number of performance measures versus number of decision making units in DEA," Annals of Operations Research, Springer, vol. 303(1), pages 529-562, August.
    18. Sabourin, Anne & Segers, Johan, 2016. "Marginal standardization of upper semicontinuous processes with application to max-stable processes," LIDAM Discussion Papers ISBA 2016019, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    19. J Morio & R Pastel, 2012. "Plug-in estimation of d-dimensional density minimum volume set of a rare event in a complex system," Journal of Risk and Reliability, , vol. 226(3), pages 337-345, June.
    20. Jaume Puig, 1999. "Radial measures of public services deficit for regional allocation of public funds," Working Papers, Research Center on Health and Economics 439, Department of Economics and Business, Universitat Pompeu Fabra.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:csdana:v:77:y:2014:i:c:p:130-145. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/csda .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.