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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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    Cited by:

    1. Becquart, Colombe & Archimbaud, Aurore & Ruiz-Gazen, Anne & Prilé, Luka & Nordhausen, Klaus, 2024. "Invariant Coordinate Selection and Fisher Discriminant Subspace Beyond The Case of Two Groups," TSE Working Papers 24-1579, Toulouse School of Economics (TSE).
    2. 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.
    3. 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.
    4. Haruhiko Ogasawara, 2019. "The multiple Cantelli inequalities," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 28(3), pages 495-506, September.
    5. Ogasawara, Haruhiko, 2018. "The inverse survival function for multivariate distributions and its application to the product moment," Statistics & Probability Letters, Elsevier, vol. 142(C), pages 71-76.
    6. Nicola Loperfido, 2019. "Finite mixtures, projection pursuit and tensor rank: a triangulation," 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. 13(1), pages 145-173, March.
    7. 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.
    8. 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.
    9. Shushi, Tomer, 2019. "A note on the coefficients of elliptical random variables," Statistics & Probability Letters, Elsevier, vol. 152(C), pages 153-155.
    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, 2024. "The skewness of mean–variance normal mixtures," Journal of Multivariate Analysis, Elsevier, vol. 199(C).
    12. Loperfido, Nicola, 2014. "Linear transformations to symmetry," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 186-192.
    13. Shushi, Tomer, 2018. "Generalized skew-elliptical distributions are closed under affine transformations," Statistics & Probability Letters, Elsevier, vol. 134(C), pages 1-4.
    14. Loperfido, Nicola, 2015. "Vector-valued skewness for model-based clustering," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 230-237.
    15. Loperfido, Nicola, 2018. "Skewness-based projection pursuit: A computational approach," Computational Statistics & Data Analysis, Elsevier, vol. 120(C), pages 42-57.
    16. 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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