Recurrence relations for bivariate t and extended skew-t distributions and an application to order statistics from bivariate t

In this paper, we derive recurrence relations for cumulative distribution functions (cdf's) of bivariate t and extended skew-t distributions. These recurrence relations are over [nu] (the degrees of freedom), and starting from the known results for [nu]=1 and [nu]=2, they will allow for the recursive evaluation of the distribution function for any other positive integral value of [nu]. Then, we consider a linear combination of order statistics from a bivariate t distribution with an arbitrary mean vector and show that its cdf is a mixture of cdf's of the extended skew-t distributions. This mixture form, along with the explicit expressions of the cdf's of the extended skew-t distributions, enables us to derive explicit expressions for the cdf of the linear combination for any positive integral value of [nu].