Generalized Likelihood Ratio test for correlated data
A simple approach would be to transform your observations $\bar{z}$ into independent random variables is by using whitening transformation, i.e., if the covariance matrix $\Sigma$ is known, then find $W$ such that $W^TW = \Sigma^{-1}$. Next, define a new random variable $\bar{y}$ as $W \bar{z}$. The above transformation is only possible when $\Sigma$ is invertible.