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

Abstract

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.

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Correspondence to Božidar V. Popović.

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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). https://doi.org/10.1007/s00009-018-1201-1

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  • DOI: https://doi.org/10.1007/s00009-018-1201-1

Keywords

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

Mathematics Subject Classification

  • 60G70
  • 62G32
  • 62E15