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Efficient recursive computational algorithms for multivariate t and multivariate unified skew-t distributions with applications to inference

Author

Listed:
  • Mehdi Amiri

    (University of Hormozgan)

  • Yaser Mehrali

    (University of Khansar)

  • Narayanaswamy Balakrishnan

    (McMaster University)

  • Ahad Jamalizadeh

    (Shahid Bahonar University of Kerman)

Abstract

In this paper, we establish efficient recursive algorithms for the computation of the cumulative distribution function (cdf) of multivariate Student’s t and multivariate unified skew-t distributions. The recurrence relations are over $$\nu $$ ν (the degrees of freedom), and starting from the explicit results for $$\nu $$ ν =1 and $$\nu $$ ν =2, they enable the recursive evaluation of the cdf for any positive integral value of $$\nu $$ ν . Using these, we obtain results for the computation of orthant probabilities of multivariate Student’s t distribution. We then demonstrate the usefulness of the established results in some problems involving order statistics and reliability systems. Finally, we use two real data sets to illustrate the methods established here.

Suggested Citation

  • Mehdi Amiri & Yaser Mehrali & Narayanaswamy Balakrishnan & Ahad Jamalizadeh, 2022. "Efficient recursive computational algorithms for multivariate t and multivariate unified skew-t distributions with applications to inference," Computational Statistics, Springer, vol. 37(1), pages 125-158, March.
    • Handle: RePEc:spr:compst:v:37:y:2022:i:1:d:10.1007_s00180-021-01119-x
    DOI: 10.1007/s00180-021-01119-x
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    References listed on IDEAS

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    1. Reinaldo B. Arellano-Valle & Marc G. Genton, 2010. "Multivariate extended skew-t distributions and related families," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 201-234.
    2. Jamalizadeh, A. & Balakrishnan, N., 2010. "Distributions of order statistics and linear combinations of order statistics from an elliptical distribution as mixtures of unified skew-elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1412-1427, July.
    3. Arellano-Valle, Reinaldo B. & Genton, Marc G., 2007. "On the exact distribution of linear combinations of order statistics from dependent random variables," Journal of Multivariate Analysis, Elsevier, vol. 98(10), pages 1876-1894, November.
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    5. Adelchi Azzalini & Marc G. Genton, 2008. "Robust Likelihood Methods Based on the Skew‐t and Related Distributions," International Statistical Review, International Statistical Institute, vol. 76(1), pages 106-129, April.
    6. Kotz,Samuel & Nadarajah,Saralees, 2004. "Multivariate T-Distributions and Their Applications," Cambridge Books, Cambridge University Press, number 9780521826549.
    7. Adelchi Azzalini & Antonella Capitanio, 2003. "Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 367-389, May.
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