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A note on the fourth cumulant of a finite mixture distribution

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  • Loperfido, Nicola

Abstract

The paper shows that the fourth cumulant of a finite mixture distribution might be decomposed into the mean of the components’ fourth cumulants and the fourth cumulant of the components’ means, when the mixture’s components have the same second and third cumulants. Statistical applications include robustness properties of likelihood-based testing procedures and kurtosis-based projection methods. Practical relevance of theoretical results in the paper are illustrated with two well-known data sets.

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  • 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.
    • Handle: RePEc:eee:jmvana:v:123:y:2014:i:c:p:386-394
    DOI: 10.1016/j.jmva.2013.09.007
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    References listed on IDEAS

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    1. Matsuura, Shun & Kurata, Hiroshi, 2011. "Principal points of a multivariate mixture distribution," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 213-224, February.
    2. 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.
    3. David Brillinger, 1969. "The calculation of cumulants via conditioning," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 21(1), pages 215-218, December.
    4. 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.
    5. 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.
    6. 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.
    7. Yanagihara, Hirokazu & Tonda, Tetsuji & Matsumoto, Chieko, 2005. "The effects of nonnormality on asymptotic distributions of some likelihood ratio criteria for testing covariance structures under normal assumption," Journal of Multivariate Analysis, Elsevier, vol. 96(2), pages 237-264, October.
    8. Tzy-Chy Lin & Tsung-I Lin, 2010. "Supervised learning of multivariate skew normal mixture models with missing information," Computational Statistics, Springer, vol. 25(2), pages 183-201, June.
    9. 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.
    10. Bernardi, Mauro & Maruotti, Antonello & Petrella, Lea, 2012. "Skew mixture models for loss distributions: A Bayesian approach," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 617-623.
    11. Loperfido, Nicola, 2013. "Skewness and the linear discriminant function," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 93-99.
    12. Matsuura, Shun & Kurata, Hiroshi, 2010. "A principal subspace theorem for 2-principal points of general location mixtures of spherically symmetric distributions," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1863-1869, December.
    13. Magnus, J.R. & Neudecker, H., 1979. "The commutation matrix : Some properties and applications," Other publications TiSEM d0b1e779-7795-4676-ac98-1, Tilburg University, School of Economics and Management.
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    Cited by:

    1. Loperfido, Nicola, 2021. "Some theoretical properties of two kurtosis matrices, with application to invariant coordinate selection," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    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. Loperfido, Nicola, 2014. "Linear transformations to symmetry," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 186-192.

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