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Scale and shape mixtures of multivariate skew-normal distributions

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  • Arellano-Valle, Reinaldo B.
  • Ferreira, Clécio S.
  • Genton, Marc G.

Abstract

We introduce a broad and flexible class of multivariate distributions obtained by both scale and shape mixtures of multivariate skew-normal distributions. We present the probabilistic properties of this family of distributions in detail and lay down the theoretical foundations for subsequent inference with this model. In particular, we study linear transformations, marginal distributions, selection representations, stochastic representations and hierarchical representations. We also describe an EM-type algorithm for maximum likelihood estimation of the parameters of the model and demonstrate its implementation on a wind dataset. Our family of multivariate distributions unifies and extends many existing models of the literature that can be seen as submodels of our proposal.

Suggested Citation

  • Arellano-Valle, Reinaldo B. & Ferreira, Clécio S. & Genton, Marc G., 2018. "Scale and shape mixtures of multivariate skew-normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 98-110.
    • Handle: RePEc:eee:jmvana:v:166:y:2018:i:c:p:98-110
    DOI: 10.1016/j.jmva.2018.02.007
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    References listed on IDEAS

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    1. Arellano-Valle, Reinaldo B., 2001. "On some characterizations of spherical distributions," Statistics & Probability Letters, Elsevier, vol. 54(3), pages 227-232, October.
    2. Vilca, Filidor & Balakrishnan, N. & Zeller, Camila Borelli, 2014. "Multivariate Skew-Normal Generalized Hyperbolic distribution and its properties," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 73-85.
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    4. Kahrari, F. & Arellano-Valle, R.B. & Rezaei, M. & Yousefzadeh, F., 2017. "Scale mixtures of skew-normal-Cauchy distributions," Statistics & Probability Letters, Elsevier, vol. 126(C), pages 1-6.
    5. Arellano-Valle, Reinaldo B. & Genton, Marc G. & Loschi, Rosangela H., 2009. "Shape mixtures of multivariate skew-normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 91-101, January.
    6. Arellano-Valle, Reinaldo B. & Bolfarine, Heleno, 1995. "On some characterizations of the t-distribution," Statistics & Probability Letters, Elsevier, vol. 25(1), pages 79-85, October.
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    12. Reinaldo B. Arellano‐Valle & Adelchi Azzalini, 2006. "On the Unification of Families of Skew‐normal Distributions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(3), pages 561-574, September.
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    14. Arellano-Valle, R. B. & del Pino, G. & San Martín, E., 2002. "Definition and probabilistic properties of skew-distributions," Statistics & Probability Letters, Elsevier, vol. 58(2), pages 111-121, June.
    15. Ahad Jamalizadeh & Tsung-I Lin, 2017. "A general class of scale-shape mixtures of skew-normal distributions: properties and estimation," Computational Statistics, Springer, vol. 32(2), pages 451-474, June.
    16. A. Azzalini & A. Capitanio, 1999. "Statistical applications of the multivariate skew normal distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 579-602.
    17. Branco, Márcia D. & Dey, Dipak K., 2001. "A General Class of Multivariate Skew-Elliptical Distributions," Journal of Multivariate Analysis, Elsevier, vol. 79(1), pages 99-113, October.
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    2. Rezaei, Amir & Yousefzadeh, Fatemeh & Arellano-Valle, Reinaldo B., 2020. "Scale and shape mixtures of matrix variate extended skew normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    3. Azzalini, Adelchi, 2022. "An overview on the progeny of the skew-normal family— A personal perspective," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    4. Abbas Mahdavi & Vahid Amirzadeh & Ahad Jamalizadeh & Tsung-I Lin, 2021. "Maximum likelihood estimation for scale-shape mixtures of flexible generalized skew normal distributions via selection representation," Computational Statistics, Springer, vol. 36(3), pages 2201-2230, September.
    5. Wang, Kesen & Karling, Maicon J. & Arellano-Valle, Reinaldo B. & Genton, Marc G., 2024. "Multivariate unified skew-t distributions and their properties," Journal of Multivariate Analysis, Elsevier, vol. 203(C).
    6. Mohammadi, Raziyeh & Kazemi, Iraj, 2022. "A robust linear mixed-effects model for longitudinal data using an innovative multivariate skew-Huber distribution," Journal of Multivariate Analysis, Elsevier, vol. 187(C).
    7. Wan-Lun Wang & Ahad Jamalizadeh & Tsung-I Lin, 2020. "Finite mixtures of multivariate scale-shape mixtures of skew-normal distributions," Statistical Papers, Springer, vol. 61(6), pages 2643-2670, December.
    8. Lee, Sharon X. & McLachlan, Geoffrey J., 2022. "An overview of skew distributions in model-based clustering," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    9. Santos, Fábio Sandro dos & Nascimento, Kerolly Kedma Felix do & Jale, Jader da Silva & Stosic, Tatijana & Marinho, Manoel H.N. & Ferreira, Tiago A.E., 2021. "Mixture distribution and multifractal analysis applied to wind speed in the Brazilian Northeast region," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    10. Jaime Arrué & Reinaldo B. Arellano-Valle & Enrique Calderín-Ojeda & Osvaldo Venegas & Héctor W. Gómez, 2023. "Likelihood Based Inference and Bias Reduction in the Modified Skew-t-Normal Distribution," Mathematics, MDPI, vol. 11(15), pages 1-21, July.
    11. Xiong, Qian & Peng, Zuoxiang, 2020. "Asymptotic expansions of powered skew-normal extremes," Statistics & Probability Letters, Elsevier, vol. 158(C).
    12. Baishuai Zuo & Narayanaswamy Balakrishnan & Chuancun Yin, 2023. "An analysis of multivariate measures of skewness and kurtosis of skew-elliptical distributions," Papers 2311.18176, arXiv.org.
    13. Hok Shing Kwong & Saralees Nadarajah, 2022. "A New Robust Class of Skew Elliptical Distributions," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1669-1691, September.

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