Skip to main content

On Extremes of Two-Dimensional Student-t Distribution of the Marshall–Olkin Type


Although there are some results related to classical bivariate Student-t distribution, studying the exact distribution of its extremes is not so easy. However, the extreme values of a bivariate Student-t distribution may play an important role in both statistical theory and practice. Therefore, this manuscript represents a pioneer work related to the studying extreme values of the bivariate Student-t distribution. For this reason, we consider another two-dimensional Student-t distribution, which is defined using the Marshall–Olkin approach. The difficulty in obtaining nice expressions for the exact distribution of the extremes for bivariate Student-t distribution may be solved by studying a more friendly distribution. The Marshall–Olkin approach is a good choice since it naturally involves extremes of the random variables. Therefore, this is one of the motivation for studying bivariate Student-t distribution of the Marshall Olkin (MO) type. Then, we study the distribution of the extremes \(M=\min \{X_1,X_2\}\) and \(S=\max \{X_1,X_2\}\), where random vector \((X_1,X_2)\) is from bivariate MO Student-t distribution. We obtain the moments and compute the percentiles of the distributions.

This is a preview of subscription content, access via your institution.


  1. Afonja, B.: The moments of the maximum of correlated normal and t-variates. J. R. Stat. Soc. Ser. B 34, 251–262 (1972)

    MathSciNet  MATH  Google Scholar 

  2. Andrews, G., Askey, R., Roy, R.: Special functions. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  3. Amos, D.E., Bulgren, W.G.: On the computation of a bivariate t-distribution. Math. Comput. 23, 319–333 (1969)

    MathSciNet  MATH  Google Scholar 

  4. Barreto-Souza, W., Lemonte, A.J.: Bivariate Kumaraswamy distribution: properties and a new method to generate bivariate classes. Statistics 47, 1321–1342 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  5. Cain, M.: Forecasting with the maximum of correlated components having bivariate t-distributed errors. IMA J. Math. Appl. Bus. Ind. 7, 233–237 (1996)

    MATH  Google Scholar 

  6. Cassidy, D.T.: A multivariate Student’s \(t\)-distribution. Open J. Stat. 6, 443–450 (2016)

    Article  Google Scholar 

  7. Dunnett, C.W., Sobel, M.: A bivariate generalization of Student’s t-distribution, with tables for certain special cases. Biometrika 41, 153–169 (1954)

    MathSciNet  Article  MATH  Google Scholar 

  8. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Academic Press, San Diego (2007)

    MATH  Google Scholar 

  9. Hakamipour, N., Mohammadpour, A., Nadarajah, S.: Extremes of bivariate Pareto distribution. Inf. Technol. Control 40, 83–87 (2011)

    Google Scholar 

  10. Heracleous, M.S.: Volatility Modeling Using the Student’s t Distribution. PhD dissertation, Faculty of the Virginia Polytechnic Institute and State University (2003)

  11. Jamalizadeh, A., Khosravi, M., Balakrishnan, N.: Recurrence relations for distributions of a skew-t and a linear combination of order statistics from a bivariate-t. Comput. Stat. Data Anal. 53, 847–852 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  12. Jamalizadeh, A., Kundu, D.: Weighted Marshall–Olkin bivariate exponential distribution. Statistics 47, 917–928 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  13. Ker, A.: On the maximum of bivariate normal random variables. Extremes 4, 185–190 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  14. Kotz, S., Nadarajah, S.: Multivariate t Distributions and Their Applications. Cambridge University Press, New York (2004)

    Book  MATH  Google Scholar 

  15. Lange, K.L., Roderick, J.A.L., Jeremy, M.G.: Robust statistical modeling using the t distribution. J. Am. Stat. Assoc. 84, 881–896 (1989)

    MathSciNet  Google Scholar 

  16. Leung, W.Y.J., Choy, S.T.B.: Robustness in forecasting future liabilities in insurance. In: Kreinovich, V., Sriboonchitta, S., Huynh, V. (eds.) Robustness in Econometrics. Springer, Berlin (2017)

    Google Scholar 

  17. Lien, D.: On the minimum and maximum of bivariate lognormal random variables. Extremes 8, 79–83 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  18. Lin, E.: Some characterization of multivariate \(t\) distribution. J. Multivar. Anal. 2, 339–344 (1972)

    MathSciNet  Article  Google Scholar 

  19. Lye, J.N., Martin, V.L.: Robust estimation, nonnormalities, and generalized exponential distributions. J. Am. Stat. Assoc. 88, 261–267 (1993)

    MATH  Google Scholar 

  20. Marshall, A.W., Olkin, I.: A multivariate exponential distribution. J. Am. Stat. Assoc. 62, 30–44 (1967)

    MathSciNet  Article  MATH  Google Scholar 

  21. Marshall, A.W., Olkin, I.: A bivariate Gompertz–Makeham life distribution. J. Multivar. Anal. 139, 219–226 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  22. Nadarajah, S., Kotz, S.: Multitude of bivariate t distributions. Statistics 38, 527–539 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  23. Roth, M.: On the Multivariate \(t\) Distribution. Technical report from Automatic Control at Linkopings universitet, Report no.: LiTH-ISY-R-3059 (2013)

  24. Reiss, R., Thomas, M.: Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields, 3rd edn. Birkhauser Verlag, Basel–Boston–Berlin (2007)

  25. Sarhan, A.M., Balakrishnan, N.: A new class of bivariate distributions and its mixture. J. Multivar. Anal. 98, 1508–1527 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  26. Siddiqui, M.: A Bivariate \(t\) distribution. Ann. Math. Stat. 38(1), 162–166 (1967)

    Article  MATH  Google Scholar 

  27. Shaw, W.T., Lee, K.T.A.: Bivariate Student \(t\) distributions with variable marginal degrees of freedom and independence. J. Multivar. Anal. 99, 1276–1287 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  28. Sweeting, P.: Financial Enterprise Risk Management. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  29. Wolfram Research, Inc.: Mathematica, Version 9.0, Champaign, IL (2012)

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Božidar V. Popović.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Popović, B.V., Genç, A.İ. On Extremes of Two-Dimensional Student-t Distribution of the Marshall–Olkin Type. Mediterr. J. Math. 15, 153 (2018).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI:


  • Extreme values
  • Marshall–Olkin-type bivariate Student-t distribution
  • moments

Mathematics Subject Classification

  • 60G70
  • 62G32
  • 62E15