Alternative Parameterization

Consider the alternative MA parameters

$\displaystyle \tilde{\mbox{\boldmath $\theta$}} = [ \tilde{b}_0, \tilde{b}_1, \ldots \tilde{b}_Q],$

where $ \tilde{b}_0$ is not constrained to be 1 and we assume that $ e_t$ has unit variance ( $ \sigma^2=1$). We can relate the parameterization $ \tilde{\mbox{\boldmath $\theta$}}$ to $ \theta$ by a 1:1 transformation

$\displaystyle \sigma^2=\tilde{b}_0^2, \;\; b_i = \tilde{b}_i/\tilde{b}_0, \; 1\leq i \leq Q.$

The Fisher's information can be transformed using the matrix of derivatives

$\displaystyle {\bf H}_{ij} = \left[{\partial \theta_i \over \partial \tilde{\th...
...0 \\
-\tilde{b}_3/\tilde{b}_0^2 & 0 & 0 & 1/\tilde{b}_0
\end{array}\right]
$

We have that

$\displaystyle \left[ \frac{ \partial \log p({\bf x}) } { \partial \tilde{\theta...
...\bf H}\;
\left[ \frac{ \partial \log p({\bf x}) } { \partial \theta } \right],$

and

$\displaystyle {\bf I}_{\tilde{\mbox{\boldmath $\theta$}}} = {\bf H}^\prime \; {\bf I}_{\mbox{\boldmath $\theta$}} \; {\bf H}.$



Baggenstoss 2017-05-19