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

In this paper, we derive recurrence relations for cumulative distribution functions (cdf’s) of bivariate tt and extended skew-tt distributions. These recurrence relations are over νν (the degrees of freedom), and starting from the known results for ν=1ν=1 and ν=2ν=2, they will allow for the recursive evaluation of the distribution function for any other positive integral value of νν. Then, we consider a linear combination of order statistics from a bivariate tt distribution with an arbitrary mean vector and show that its cdf is a mixture of cdf’s of the extended skew-tt distributions. This mixture form, along with the explicit expressions of the cdf’s of the extended skew-tt distributions, enables us to derive explicit expressions for the cdf of the linear combination for any positive integral value of νν.