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 \(\nu \) (the degrees of freedom), and starting from the explicit results for \(\nu \)=1 and \(\nu \)=2, they enable the recursive evaluation of the cdf for any positive integral value of \(\nu \). 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.
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References
Adcock C, Eling M, Loperfido N (2015) Skewed distributions in finance and actuarial science: a review. The European Journal of Finance 21(13–14):1253–1281
Arellano-Valle RB, Genton MG (2007) On the exact distribution of linear combinations of order statistics from dependent random variables. Journal of Multivariate Analysis 98(10):1876–1894
Arellano-Valle RB, Genton MG (2010) Multivariate extended skew-\(t\) distributions and related families. Metron 68(3):201–234
Azzalini A, Capitanio A (2003) Distributions generated by perturbation of symmetry with emphasis on a multivariate skew \(t\) distribution. Journal of the Royal Statistical Society, Series B 65(2):367–389
Azzalini A, Dalla-Valle A (1996) The multivariate skew-normal distribution. Biometrika 83(4):715–726
Azzalini A, Genton MG (2008) Robust likelihood methods based on the skew-t and related distributions. International Statistical Review 76(1):106–129
Bacon RH (1963) Approximations to multivariate normal orthant probabilities. The Annals of Mathematical Statistics 34(1):191–198
Cook RD, Johnson ME (1986) Generalized Burr Pareto logistic distributions with applications to a uranium exploration data set. Technometrics 28:123–131
Fang K, Kotz S, Ng KW (1990) Symmetric Multivariate and Related Distributions. Chapman & Hall, London
González-Farías, G., Domínquez-Molina, A. and Gupta A.K.(2004). Additive properties of skew normal random vectors. Journal of Statistical Planning and Inference, 126(2), 521-534
González-Farías, G., Domínquez-Molina, A. and Gupta A.K.(2004). The closed skew-normal distribution. In: Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality, Genton, M. G., Ed., Chapman & Hall / CRC, Boca Raton, FL, pp. 25-42
Jamalizadeh A, Balakrishnan N (2009) Order statistics from trivariate normal and \(t_{\nu }\)-distributions in terms of generalized skew-normal and skew-\(t_{\nu }\) distributions. Journal of Statistical Planning and Inference 139(11):3799–3819
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 101(6):1412–1427
Jamalizadeh A, Balakrishnan N (2012) Concomitants of order statistics from multivariate elliptical distributions. Journal of Statistical Planning and Inference 142(2):397–409
Jamalizadeh A, Khosravi M, Balakrishnan N (2009a) Recurrence relations for distributions of a skew-\(t\) and a linear combination of order statistics from a bivariate-\(t\). Computational Statistics & Data Analysis 53(4):847–852
Jamalizadeh A, Mehrali Y, Balakrishnan N (2009b) Recurrence relations for bivariate \(t\) and extended skew-\(t\) distributions and an application to order statistics from bivariate \(t\). Computational Statistics & Data Analysis 53(12):4018–4027
Kalbfleisch JD, Prentice RL (2002) The Statistical Analysis of Failure Time Data, 2nd edn. John Wiley & Sons, New York
Kotz S, Balakrishnan N, Johnson NL (2000) Continuous Multivariate Distributions-Vol. 1, 2nd edn. John Wiley & Sons, New York
Kotz S, Nadarajah S (2004) Multivariate t-distributions and Their Applications. Cambridge University Press, Cambridge, England
Loperfido N, Navarro J, Ruiz JM, Sandoval CJ (2007) Some relationships between skew-normal distributions and order statistics from exchangeable normal random vectors. Communications in Statistics- Theory and Methods 36:1719–1733
Kelkin Nama M, Asadi M (2014) Stochastic properties of components in a used coherent system. Methodology and Computing in Applied Probability 16(3):675–691
Tong, Y.L. (2012). The Multivariate Normal Distribution, Springer, New York.s
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The authors express their sincere thanks to the Editor and the anonymous reviewers for all their useful comments and suggestions on an earlier version of this manuscript which led to this improved version.
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Amiri, M., Mehrali, Y., Balakrishnan, N. et al. Efficient recursive computational algorithms for multivariate t and multivariate unified skew-t distributions with applications to inference. Comput Stat 37, 125–158 (2022). https://doi.org/10.1007/s00180-021-01119-x
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DOI: https://doi.org/10.1007/s00180-021-01119-x