I am trying to find closed form solutions to the following maximum likelihood optimization:
$$\sum_{i=1}^{a}\sum_{j=1}^{b}\sum_{k=1}^{n_{ij}}(y_{ijk}-\mu-\alpha_i-\eta_j-\gamma_{ij})^2$$
I tried to find the solution by taking the derivative with respect to each parameter, but I could not find a closed solution for the estimators $\hat{\alpha_i}$, $\hat{\eta_j}$, $\hat{\mu}$,$\hat{\gamma_{ij}}$
I believe this is an overparametrization issue so I need to impose some constraints on the parameters to find a closed form solution. Is this correct or is there a way to find solutions?