I have two multivariate normal populations: $\boldsymbol{A_1}$~$N_r(\boldsymbol{\mu_1},\Sigma)$ with $n_1$ observations, and $\boldsymbol{A_2}$~$N_r(\boldsymbol{\mu_2},\Sigma)$ with $n_2$ observations. I am wondering if there is some way to express the likelihood of $\boldsymbol{A_1}-\boldsymbol{A_2}$. Separately, I know the likelihoods of $\boldsymbol{A_1}$ and $\boldsymbol{A_2}$ are: $L(\boldsymbol{A_1})=L(\boldsymbol{\mu_1},\Sigma)=(2\pi)^{-\frac{1}{2}pn_1}|\Sigma|^{-\frac{1}{2}n_1}exp{[-\frac{1}{2}\sum_{\alpha=1}^{n_1}(\boldsymbol{A_{1,\alpha}-\mu_1})^T\Sigma^{-1}(\boldsymbol{A_{1,\alpha}-\mu_1})]}$ and $L(\boldsymbol{A_2})=L(\boldsymbol{\mu_2},\Sigma)=(2\pi)^{-\frac{1}{2}pn_2}|\Sigma|^{-\frac{1}{2}n_2}exp{[-\frac{1}{2}\sum_{\alpha=1}^{n_2}(\boldsymbol{A_{2,\alpha}-\mu_2})^T\Sigma^{-1}(\boldsymbol{A_{2,\alpha}-\mu_2})]}$, respectively. The end goal is to find the MLEs of $\boldsymbol{A_1}-\boldsymbol{A_2}$ (both the … Read more