We now discuss parameter orthogonality, which has important consequences for likelihood characteristics and inference.
Let of dimension and respectively , i.e. and . The expected information matrix, , can be partitioned as follows
where has th element
similarly has th element
and has th element
Definition of Orthogonality
We define to be orthogonal
to if , i.e.
the elements of this matrix satisfy the property
for and whatever value of in .
Closely linked to this is the orthogonality of the observed
information matrix at .
Orthogonality of ( means that the maximum likelihood estimates and are asymptotically independent. This has a number of practical advantages:
Parameter interpretation;
The asymptotic standard error for estimating is the same whether is treated as known or unknown;
Likelihood based confidence intervals are simpler to summarize and interpret.