IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i16p3579-d1220073.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/16/3579/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/11/16/3579/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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. Saralees Nadarajah, 2007. "A truncated bivariate inverted dirichlet distribution," Statistica, Department of Statistics, University of Bologna, vol. 67(2), pages 213-221.
    6. Horrace, William C., 2005. "Some results on the multivariate truncated normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 209-221, May.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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. Badía, F.G. & Sangüesa, C. & Cha, J.H., 2014. "Stochastic comparison of multivariate conditionally dependent mixtures," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 82-94.
    3. Arismendi, J.C., 2013. "Multivariate truncated moments," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 41-75.
    4. Sharon Lee & Geoffrey McLachlan, 2013. "On mixtures of skew normal and skew $$t$$ -distributions," 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. 7(3), pages 241-266, September.
    5. Chunzheng Cao & Mengqian Chen & Yahui Wang & Jian Qing Shi, 2018. "Heteroscedastic replicated measurement error models under asymmetric heavy-tailed distributions," Computational Statistics, Springer, vol. 33(1), pages 319-338, March.
    6. C. Adcock, 2010. "Asset pricing and portfolio selection based on the multivariate extended skew-Student-t distribution," Annals of Operations Research, Springer, vol. 176(1), pages 221-234, April.
    7. Chunzheng Cao & Yahui Wang & Jian Qing Shi & Jinguan Lin, 2018. "Measurement Error Models for Replicated Data Under Asymmetric Heavy-Tailed Distributions," Computational Economics, Springer;Society for Computational Economics, vol. 52(2), pages 531-553, August.
    8. Roozegar, Roohollah & Balakrishnan, Narayanaswamy & Jamalizadeh, Ahad, 2020. "On moments of doubly truncated multivariate normal mean–variance mixture distributions with application to multivariate tail conditional expectation," Journal of Multivariate Analysis, Elsevier, vol. 177(C).
    9. Azzalini, Adelchi, 2022. "An overview on the progeny of the skew-normal family— A personal perspective," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    10. Oleg Badunenko & Daniel J. Henderson, 2024. "Production analysis with asymmetric noise," Journal of Productivity Analysis, Springer, vol. 61(1), pages 1-18, February.
    11. 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.
    12. Cabral, Celso Rômulo Barbosa & Lachos, Víctor Hugo & Prates, Marcos O., 2012. "Multivariate mixture modeling using skew-normal independent distributions," Computational Statistics & Data Analysis, Elsevier, vol. 56(1), pages 126-142, January.
    13. Sharon Lee & Geoffrey McLachlan, 2013. "Model-based clustering and classification with non-normal mixture distributions," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 22(4), pages 427-454, November.
    14. Lachos, Victor H. & Bandyopadhyay, Dipankar & Garay, Aldo M., 2011. "Heteroscedastic nonlinear regression models based on scale mixtures of skew-normal distributions," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1208-1217, August.
    15. 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.
    16. Adcock, C.J., 2014. "Mean–variance–skewness efficient surfaces, Stein’s lemma and the multivariate extended skew-Student distribution," European Journal of Operational Research, Elsevier, vol. 234(2), pages 392-401.
    17. Alfonso Flores-Lagunes & William C. Horrace & Kurt E. Schnier, 2007. "Identifying technically efficient fishing vessels: a non-empty, minimal subset approach," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 22(4), pages 729-745.
    18. Roberto Colombi & Subal Kumbhakar & Gianmaria Martini & Giorgio Vittadini, 2014. "Closed-skew normality in stochastic frontiers with individual effects and long/short-run efficiency," Journal of Productivity Analysis, Springer, vol. 42(2), pages 123-136, October.
    19. Panagiotelis, Anastasios & Smith, Michael, 2010. "Bayesian skew selection for multivariate models," Computational Statistics & Data Analysis, Elsevier, vol. 54(7), pages 1824-1839, July.
    20. Lin, Edward M.H. & Sun, Edward W. & Yu, Min-Teh, 2020. "Behavioral data-driven analysis with Bayesian method for risk management of financial services," International Journal of Production Economics, Elsevier, vol. 228(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:16:p:3579-:d:1220073. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.