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

Jones, M. C. (2002). A dependent bivariate t distribution with marginals on different degrees of freedom. Statistics & Probability Letters, 56(2) pp. 163–170.

DOI: https://doi.org/10.1016/S0167-7152(01)00180-8

Abstract

Let Z1,Z2 and W1,W2 be mutually independent random variables, each Zi following the standard normal distribution and Wi following the chi-squared distribution on ni degrees of freedom. Then, the pair of random variables {√n1Z1/√W1, √n1Z2/√W1} has the bivariate spherically symmetric t distribution; this has both marginals the same, namely Student's t distributions on n1 degrees of freedom. In this paper, we study the joint distribution of {√ν1Z1/√W1, √ν2Z2/√W1+W2} where ν1=n1, ν2=n1+n2. This bivariate distribution has marginal distributions which are Student t distributions on different degrees of freedom if ν1≠ν2. The marginals remain uncorrelated, as in the spherically symmetric case, but are also by no means independent.

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