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Skewness and the linear discriminant function

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  • Loperfido, Nicola

Abstract

Fisher’s linear discriminant function might be difficult to estimate, when data come from a semiparametric, finite mixture model. We propose an estimator based on the singular value decomposition of the third standardized cumulant. The estimator is consistent when sampling from a mixture of two symmetric, homoscedastic components with finite third moments and different weights. We also evaluate its performance and compare it with another estimator, which uses the eigenvectors of a kurtosis matrix.

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  • Loperfido, Nicola, 2013. "Skewness and the linear discriminant function," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 93-99.
    • Handle: RePEc:eee:stapro:v:83:y:2013:i:1:p:93-99
    DOI: 10.1016/j.spl.2012.08.032
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    References listed on IDEAS

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    1. Kollo, Tõnu, 2008. "Multivariate skewness and kurtosis measures with an application in ICA," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2328-2338, November.
    2. T. P. Hettmansperger & Hoben Thomas, 2000. "Almost nonparametric inference for repeated measures in mixture models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(4), pages 811-825.
    3. Peña, Daniel & Prieto, Francisco J. & Viladomat, Júlia, 2010. "Eigenvectors of a kurtosis matrix as interesting directions to reveal cluster structure," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 1995-2007, October.
    4. Peña, Daniel & Prieto, Francisco J., 2000. "The kurtosis coefficient and the linear discriminant function," Statistics & Probability Letters, Elsevier, vol. 49(3), pages 257-261, September.
    5. Marc Genton & Nicola Loperfido, 2005. "Generalized skew-elliptical distributions and their quadratic forms," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(2), pages 389-401, June.
    6. Pena D. & Prieto F.J., 2001. "Cluster Identification Using Projections," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1433-1445, December.
    7. Giovanni De Luca & Nicola Loperfido, 2015. "Modelling multivariate skewness in financial returns: a SGARCH approach," The European Journal of Finance, Taylor & Francis Journals, vol. 21(13-14), pages 1113-1131, November.
    8. Posse, Christian, 1995. "Projection pursuit exploratory data analysis," Computational Statistics & Data Analysis, Elsevier, vol. 20(6), pages 669-687, December.
    9. Adelchi Azzalini & Antonella Capitanio, 2003. "Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 367-389, May.
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    Cited by:

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    3. Haruhiko Ogasawara, 2019. "The multiple Cantelli inequalities," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 28(3), pages 495-506, September.
    4. 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.
    5. Shushi, Tomer, 2018. "Generalized skew-elliptical distributions are closed under affine transformations," Statistics & Probability Letters, Elsevier, vol. 134(C), pages 1-4.
    6. Shushi, Tomer, 2018. "A proof for the existence of multivariate singular generalized skew-elliptical density functions," Statistics & Probability Letters, Elsevier, vol. 141(C), pages 50-55.
    7. Shushi, Tomer, 2019. "A note on the coefficients of elliptical random variables," Statistics & Probability Letters, Elsevier, vol. 152(C), pages 153-155.
    8. Loperfido, Nicola, 2015. "Vector-valued skewness for model-based clustering," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 230-237.
    9. Loperfido, Nicola, 2014. "A note on the fourth cumulant of a finite mixture distribution," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 386-394.
    10. Song, Pingfan & Tan, Changchun & Wang, Shaochen, 2019. "On the moment generating function for random vectors via inverse survival function," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 345-350.
    11. Loperfido, Nicola, 2014. "Linear transformations to symmetry," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 186-192.
    12. Loperfido, Nicola, 2018. "Skewness-based projection pursuit: A computational approach," Computational Statistics & Data Analysis, Elsevier, vol. 120(C), pages 42-57.
    13. Tarpey, Thaddeus & Loperfido, Nicola, 2015. "Self-consistency and a generalized principal subspace theorem," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 27-37.
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