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A study of the effect of kurtosis on discriminant analysis under elliptical populations

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  • Arevalillo, Jorge M.
  • Navarro, Hilario

Abstract

This paper is concerned with the role some parameters indexing four important families within the multivariate elliptically contoured distributions play as indicators of multivariate kurtosis. The problem is addressed for the exponential power family, for a subclass of the Kotz family and for the Pearson type II and type VII distributions. Once such a problem is analyzed, we study the effect these parameters have, as kurtosis indicators, on binary discriminant analysis by exploring their relationship with the error rate of the Bayes discriminant rule. The effect is analyzed under mild conditions on the kernel function generating the elliptical density. Some numerical examples are given in order to illustrate our theoretical insights and findings.

Suggested Citation

  • Arevalillo, Jorge M. & Navarro, Hilario, 2012. "A study of the effect of kurtosis on discriminant analysis under elliptical populations," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 53-63.
    • Handle: RePEc:eee:jmvana:v:107:y:2012:i:c:p:53-63
    DOI: 10.1016/j.jmva.2012.01.011
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    References listed on IDEAS

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    1. Samuel Kotz & Edith Seier, 2009. "An analysis of quantile measures of kurtosis: center and tails," Statistical Papers, Springer, vol. 50(3), pages 553-568, June.
    2. Oja, Hannu, 1983. "Descriptive statistics for multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 1(6), pages 327-332, October.
    3. Wang, Jin, 2009. "A family of kurtosis orderings for multivariate distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(3), pages 509-517, March.
    4. Cambanis, Stamatis & Huang, Steel & Simons, Gordon, 1981. "On the theory of elliptically contoured distributions," Journal of Multivariate Analysis, Elsevier, vol. 11(3), pages 368-385, September.
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    Cited by:

    1. Jorge M. Arevalillo & Hilario Navarro, 2019. "A stochastic ordering based on the canonical transformation of skew-normal vectors," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 475-498, June.
    2. Jorge M. Arevalillo & Hilario Navarro, 2021. "Skewness-Kurtosis Model-Based Projection Pursuit with Application to Summarizing Gene Expression Data," Mathematics, MDPI, vol. 9(9), pages 1-18, April.
    3. Javed, Farrukh & Loperfido, Nicola & Mazur, Stepan, 2020. "Edgeworth Expansions for Multivariate Random Sums," Working Papers 2020:9, Örebro University, School of Business.
    4. Jin Wang & Weihua Zhou, 2015. "Effect of kurtosis on efficiency of some multivariate medians," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 27(3), pages 331-348, September.
    5. Jorge M. Arevalillo & Hilario Navarro, 2020. "Data projections by skewness maximization under scale mixtures of skew-normal vectors," 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. 14(2), pages 435-461, June.
    6. Nicola Loperfido, 2023. "Kurtosis removal for data pre-processing," 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. 17(1), pages 239-267, March.
    7. Arevalillo, Jorge M. & Navarro, Hilario, 2015. "A note on the direction maximizing skewness in multivariate skew-t vectors," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 328-332.

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