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.