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 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 {√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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