Chapter 3

Exercise 3.11. Show that the solution to the multivariate linear regression problem

\[\text{RSS}(\mathbf{B}; \mathbf{\Sigma}) = \sum_{i=1}^N (y_i - f(x_i))^\top \mathbf{\Sigma}^{-1}(y_i - f(x_i))\]

is given by

\[\hat{\mathbf{B}} = (\mathbf{X}^\top \mathbf{X})^{-1}\mathbf{X^\top Y}\]

What happens if the covariance matrices \(\mathbf{\Sigma}_i\) are different for each observation?