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

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  • 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
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    References listed on IDEAS

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    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.
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    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.
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