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The skewness of mean–variance normal mixtures

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  • Loperfido, Nicola

Abstract

Mean–variance mixtures of normal distributions are very flexible: they model many nonnormal features, such as skewness, kurtosis and multimodality. Special cases include generalized asymmetric Laplace distributions, mixtures of two normal distributions with proportional covariance matrices, scale mixtures of normal distributions and normal distributions. This paper investigates the skewness of multivariate mean–variance normal mixtures. The special case of mixtures of two normal distributions with proportional covariance matrices is treated in greater detail. The paper derives the analytical forms of prominent measures of multivariate skewness and applies them to model-based clustering, normalizing linear transformations, projection pursuit and normality testing. The practical relevance of the theoretical results is assessed with both real and simulated data.

Suggested Citation

  • Loperfido, Nicola, 2024. "The skewness of mean–variance normal mixtures," Journal of Multivariate Analysis, Elsevier, vol. 199(C).
    • Handle: RePEc:eee:jmvana:v:199:y:2024:i:c:s0047259x2300088x
    DOI: 10.1016/j.jmva.2023.105242
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    1. Eric Jondeau & Michael Rockinger, 2006. "Optimal Portfolio Allocation under Higher Moments," European Financial Management, European Financial Management Association, vol. 12(1), pages 29-55, January.
    2. 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.
    3. Henze, Norbert, 1997. "Limit laws for multivariate skewness in the sense of Móri, Rohatgi and Székely," Statistics & Probability Letters, Elsevier, vol. 33(3), pages 299-307, May.
    4. Loperfido, Nicola, 2021. "Some theoretical properties of two kurtosis matrices, with application to invariant coordinate selection," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    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. Pauliina Ilmonen & Hannu Oja & Robert Serfling, 2012. "On Invariant Coordinate System (ICS) Functionals," International Statistical Review, International Statistical Institute, vol. 80(1), pages 93-110, April.
    7. Loperfido, Nicola, 2015. "Vector-valued skewness for model-based clustering," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 230-237.
    8. Loperfido, Nicola, 2018. "Skewness-based projection pursuit: A computational approach," Computational Statistics & Data Analysis, Elsevier, vol. 120(C), pages 42-57.
    9. Loperfido, Nicola, 2013. "Skewness and the linear discriminant function," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 93-99.
    10. Ab Mooijaart, 1985. "Factor analysis for non-normal variables," Psychometrika, Springer;The Psychometric Society, vol. 50(3), pages 323-342, September.
    11. Agnan Kessy & Alex Lewin & Korbinian Strimmer, 2018. "Optimal Whitening and Decorrelation," The American Statistician, Taylor & Francis Journals, vol. 72(4), pages 309-314, October.
    12. 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.
    13. Tõnu Kollo & Dietrich Von Rosen, 1998. "A Unified Approach to the Approximation of Multivariate Densities," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 25(1), pages 93-109, March.
    14. N. H. Bingham & Rudiger Kiesel, 2002. "Semi-parametric modelling in finance: theoretical foundations," Quantitative Finance, Taylor & Francis Journals, vol. 2(4), pages 241-250.
    15. 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.
    16. Arnold, Barry C. & Beaver, Robert J., 2000. "The skew-Cauchy distribution," Statistics & Probability Letters, Elsevier, vol. 49(3), pages 285-290, September.
    17. 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.
    18. 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.
    19. Loperfido, Nicola, 2014. "Linear transformations to symmetry," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 186-192.
    20. Nicola Loperfido & Stepan Mazur & Krzysztof Podgórski, 2018. "Third cumulant for multivariate aggregate claim models," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2018(2), pages 109-128, February.
    21. Karlsson, Sune & Mazur, Stepan & Nguyen, Hoang, 2023. "Vector autoregression models with skewness and heavy tails," Journal of Economic Dynamics and Control, Elsevier, vol. 146(C).
    22. Genton, Marc G. & He, Li & Liu, Xiangwei, 2001. "Moments of skew-normal random vectors and their quadratic forms," Statistics & Probability Letters, Elsevier, vol. 51(4), pages 319-325, February.
    23. Wraith, Darren & Forbes, Florence, 2015. "Location and scale mixtures of Gaussians with flexible tail behaviour: Properties, inference and application to multivariate clustering," Computational Statistics & Data Analysis, Elsevier, vol. 90(C), pages 61-73.
    24. Kozubowski, Tomasz J. & Podgórski, Krzysztof & Rychlik, Igor, 2013. "Multivariate generalized Laplace distribution and related random fields," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 59-72.
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