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Suppose $(Y_1,Y_2)^T \sim N((0,0)^T,\Sigma)$, where $$ \Sigma = \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{pmatrix}. $$ $T_1= \frac{\bar{Y_1}}{S_1/\sqrt(1/n)}$ and $T_2= \frac{\bar{Y_2}}{S_2/\sqrt(1/n)}$ are the T test statistics. If $\sigma_1 = \sigma_2$ and $S_1=S_2$, then the joint distribution of $T_1, T_2$ is multivariate t distribution with the shape matrix equal to the correlation matrix corresponding to $\Sigma$ and $df=2n-2$. What is the joint distribution of $T_1, T_2$ if $\sigma_1 \ne \sigma_2$ ? Can I estimate it by the multivariate t distribution with shape matrix equal to the correlation matrix corresponding to $\Sigma$ and $df=n-1$ ?

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