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On the Asymptotics of Trimmed Best k-Nets

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  • Cuesta-Albertos, J. A.
  • García-Escudero, L. A.
  • Gordaliza, A.

Abstract

Trimmed best k-nets were introduced in J. A. Cuesta-Albertos, A. Gordaliza and C. Matrán (1998, Statist. Probab. Lett.36, 401-413) as a robustified L[infinity]-based quantization procedure. This paper focuses on the asymptotics of this procedure. Also, some possible applications are briefly sketched to motivate the interest of this technique. Consistency and weak limit law are obtained in the multivariate setting. Consistency holds for absolutely continuous distributions without the (artificial) requirement of a trimming level varying with the sample size as in J. A. Cuesta-Albertos, A. Gordaliza and C. Matrán (1998, Statist. Probab. Lett.36, 401-413). The weak convergence will be stated toward a non-normal limit law at a OP(n-1/3) rate of convergence. An algorithm for computing trimmed best k-nets is proposed. Also a procedure is given in order to choose an appropriate number of centers, k, for a given data set.

Suggested Citation

  • Cuesta-Albertos, J. A. & García-Escudero, L. A. & Gordaliza, A., 2002. "On the Asymptotics of Trimmed Best k-Nets," Journal of Multivariate Analysis, Elsevier, vol. 82(2), pages 486-516, August.
    • Handle: RePEc:eee:jmvana:v:82:y:2002:i:2:p:486-516
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    References listed on IDEAS

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    1. Li, Luning & Flury, Bernard, 1995. "Uniqueness of principal points for univariate distributions," Statistics & Probability Letters, Elsevier, vol. 25(4), pages 323-327, December.
    2. William H. Rogers & John W. Tukey, 1972. "Understanding some long‐tailed symmetrical distributions," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 26(3), pages 211-226, September.
    3. Nolan, D., 1992. "Asymptotics for multivariate trimming," Stochastic Processes and their Applications, Elsevier, vol. 42(1), pages 157-169, August.
    4. Serinko, Regis J. & Babu, Gutti Jogesh, 1992. "Weak limit theorems for univariate k-mean clustering under a nonregular condition," Journal of Multivariate Analysis, Elsevier, vol. 41(2), pages 273-296, May.
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    Cited by:

    1. Luis García-Escudero & Alfonso Gordaliza & Carlos Matrán & Agustín Mayo-Iscar, 2010. "A review of robust clustering methods," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 4(2), pages 89-109, September.

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