Find the maximum likelihood estimates of $\mu$ and $\Sigma$ in the constrained case.
The testing problem is:
$H_0 :\Sigma _{12}=0$ versus $H_1 :\Sigma _{12} \not= 0$
$X_1 \sim N( \left[ {\begin{array}{cc} \mu ^{(1)} \\ \mu ^{(2)} \\ \end{array} } \right], \left[ {\begin{array}{cc} \Sigma _{11} & \Sigma _{12} \\ \Sigma _{21} & \Sigma_{22} \\ \end{array} } \right])$
$\Sigma =\frac{1}{n}A$
Assume that the parameters $\mu$ and $\Sigma $ are unknow.
I came until $$f_X(x) =\frac{\exp\left(-\frac12(x^{(1)}-\mu ^{(1)})^T\Sigma_{11}^{-1}(x^{(1)}-\mu ^{(1)})-\frac{1}{2}(x^{(2)}-\mu ^{(2)})^T\Sigma_{22}^{-1}(x^{(2)}-\mu ^{(2)})\right)}{(2\pi)^{n/2}\sqrt{\det\Sigma_{11} \det\Sigma_{22}} } $$
Where $$S = \frac{1}{n}\sum_{k=1}^n(x_k - \bar{x})(x_k -\bar{x})^T =\left[ {\begin{array}{cc} S_{11} & S_{12} \\ S_{21} & S_{22} \\ \end{array} } \right] \qquad \hat{\mu} =\frac{1}{n}\sum_{k=1}^nx_k =\left[ {\begin{array}{cc} \bar{x} ^{(1)} \\ \bar{x}^{(2)} \\ \end{array} } \right]$$
Now I need to maximize $f_X(x)$ but I do not know how. Thank you for any help.