User:AndiWangIE/sandbox

Another special case is that the mean and covariance depends on two different parameters, say, \beta and \theta. This is especially popular in the analysis of spacial data, which uses a linear model with correlated residuals. In this case, we have

$$\mathcal{I}\left( \beta ,\theta \right)=\text{diag}\left( \mathcal{I}\left( \beta  \right),\mathcal{I}\left( \theta  \right) \right)$$

where

$$\mathcal{I}{{\left( \beta \right)}_{m,n}}=\frac{\partial {{\mu }^{\text{T}}}}{\partial {{\beta }_{m}}}{{\Sigma }^{-1}}\frac{\partial \mu }{\partial {{\beta }_{n}}}$$

and

$$\mathcal{I}{{\left( \theta \right)}_{m,n}}=\frac{1}{2}\operatorname{tr}\left( {{\Sigma }^{-1}}\frac{\partial \Sigma }{\partial {{\theta }_{m}}}{{\Sigma }^{-1}}\frac{\partial \Sigma }{\partial {{\theta }_{n}}} \right)$$

The prove of this special case is given in literature Maximum likelihood estimation of models for residual covariance in spatial regression, K.V.Mardia and R.J.Marshall, Biometrika (1984), 71, 1, pp. 135-46 <\ref>. Using the same technique in this paper, it's not difficult to prove the original result.