A dependent bivariate t distribution with marginals on different degrees of freedom

Let Z(1), Z(2) and W-1, W-2 be mutually independent random variables, each Z(i) following the standard normal distribution and W-i following the chi-squared distribution on n(i) degrees of freedom. Then, the pair of random variables {rootn(1)Z(1)/rootW(1), rootn(1)Z(2)/rootW(1)} has the bivariate spherically symmetric t distribution; this has both marginals the same, namely Student's t distributions on n(1) degrees of freedom. In this paper, we study the joint distribution of {rootnu(1)Z(1)/rootW(1), rootnu(2)Z(2)/rootW(1)+ W-2,} where v(1) = n(1), v(2) = n(1) + n(2). This bivariate distribution has marginal distributions which are Student t distributions on different degrees of freedom if nu(1) not equalnu(2). The marginals remain uncorrelated, as in the spherically symmetric case, but are also by no means independent. (C) 2002 Elsevier Science B.V. All rights reserved.