Suppose we are interested in the covariance matrix $\Sigma$ of a few MLE estimators $\hat \theta_1,\hat \theta_2,\cdots,\hat \theta_n$. For each $j$, $\hat \theta_j$ is normally distributed and estimated from data. The data is multivariate normal with known covariance and mean $\vec 0$.
The problem is, I obtained the covariance matrix $\Sigma$ heuristically because it was impossible to compute directly. Now I want to prove that I have found the correct expression. What are some methods which would prove that I have found the correct covariance matrix?
EDIT: I was asked to explain how the covariance matrices was were obtained heuristically. A few dozens of covariance matrices of interest were generated numerically. This is possible since $\hat \theta_i$ has closed form and therefore $\Sigma$ has closed form. By staring at the geometrical pattern they formed it was possible to guess the formula for the elements of the covariance matrices. Next, numerics were used again to compute thousands of such matrices to check that the guess was correct.