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Some Theoretical and Computational Aspects of the Truncated Multivariate Skew-Normal/Independent Distributions

Author

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  • Raúl Alejandro Morán-Vásquez

    (Instituto de Matemáticas, Universidad de Antioquia, Calle 67 No. 53-108, Medellín 050010, Colombia
    These authors contributed equally to this work.)

  • Edwin Zarrazola

    (Instituto de Matemáticas, Universidad de Antioquia, Calle 67 No. 53-108, Medellín 050010, Colombia
    These authors contributed equally to this work.)

  • Daya K. Nagar

    (Instituto de Matemáticas, Universidad de Antioquia, Calle 67 No. 53-108, Medellín 050010, Colombia
    These authors contributed equally to this work.)

Abstract

In this article, we derive a closed-form expression for computing the probabilities of p -dimensional rectangles by means of a multivariate skew-normal distribution. We use a stochastic representation of the multivariate skew-normal/independent distributions to derive expressions that relate their probability density functions to the expected values of positive random variables. We also obtain an analogous expression for probabilities of p -dimensional rectangles for these distributions. Based on this, we propose a procedure based on Monte Carlo integration to evaluate the probabilities of p -dimensional rectangles through multivariate skew-normal/independent distributions. We use these findings to evaluate the probability density functions of a truncated version of this class of distributions, for which we also suggest a scheme to generate random vectors by using a stochastic representation involving a truncated multivariate skew-normal random vector. Finally, we derive distributional properties involving affine transformations and marginalization. We illustrate graphically several of our methodologies and results derived in this article.

Suggested Citation

  • Raúl Alejandro Morán-Vásquez & Edwin Zarrazola & Daya K. Nagar, 2023. "Some Theoretical and Computational Aspects of the Truncated Multivariate Skew-Normal/Independent Distributions," Mathematics, MDPI, vol. 11(16), pages 1-16, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:16:p:3579-:d:1220073
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    References listed on IDEAS

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    1. Raúl Alejandro Morán-Vásquez & Edwin Zarrazola & Daya K. Nagar, 2022. "Some Statistical Aspects of the Truncated Multivariate Skew- t Distribution," Mathematics, MDPI, vol. 10(15), pages 1-14, August.
    2. Horrace, William C., 2005. "On ranking and selection from independent truncated normal distributions," Journal of Econometrics, Elsevier, vol. 126(2), pages 335-354, June.
    3. 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.
    4. Basso, Rodrigo M. & Lachos, Víctor H. & Cabral, Celso Rômulo Barbosa & Ghosh, Pulak, 2010. "Robust mixture modeling based on scale mixtures of skew-normal distributions," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 2926-2941, December.
    5. Horrace, William C., 2005. "Some results on the multivariate truncated normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 209-221, May.
    6. Saralees Nadarajah, 2007. "A truncated bivariate inverted dirichlet distribution," Statistica, Department of Statistics, University of Bologna, vol. 67(2), pages 213-221.
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