Efficient recursive computational algorithms for multivariate t and multivariate unified skew-t distributions with applications to inference

In this paper, we establish efficient recursive algorithms for the computation of the cumulative distribution function (cdf) of multivariate Student’s t and multivariate unified skew-t distributions. The recurrence relations are over ν$$\nu $$ (the degrees of freedom), and starting from the explicit results for ν$$\nu $$=1 and ν$$\nu $$=2, they enable the recursive evaluation of the cdf for any positive integral value of ν$$\nu $$. Using these, we obtain results for the computation of orthant probabilities of multivariate Student’s t distribution. We then demonstrate the usefulness of the established results in some problems involving order statistics and reliability systems. Finally, we use two real data sets to illustrate the methods established here.