ML estimation

Note that in these problems, $ \hat{\mbox{\boldmath $\theta$}}$ can be found independently of $ \hat{{\bf a}}, \; \hat{\sigma^2}$ by maximizing

$\displaystyle Q={\bf x}^\prime {\bf H}_\theta \left({\bf H}_{\hat\theta}^\prime {\bf H}_{\hat\theta}\right)^{-1}
{\bf H}_{\hat\theta}^\prime{\bf x}.$

The other parameters $ \hat{{\bf a}}$ and $ \hat{\sigma}^2$ can be obtained in closed form for a fixed $ \hat{\mbox{\boldmath $\theta$}}$,

$\displaystyle \hat{{\bf a}}=\left({\bf H}_{\hat\theta}^\prime {\bf H}_{\hat\theta}\right)^{-1}{\bf H}_{\hat\theta}^\prime{\bf x},$ (4.2)

$\displaystyle \hat{\sigma^2}=\frac{1}{N}({\bf x}-{\bf H}_{\hat\theta}\hat{{\bf a}})^\prime ({\bf x}-{\bf H}_{\hat\theta}\hat{{\bf a}}).$ (4.3)

Alternatively, the Newton-Raphson algorithm may be used to find $ \hat{\mbox{\boldmath $\theta$}}$ as described in Section 16.5. For this we need the first derivatives and Fisher's information (See Section 16.5).


Baggenstoss 2017-05-19