Under certain conditions, the J-function is independent of as long
as remains within the ``region of sufficiency" (ROS) for
(See Section 2.3.3).
Then, can even ``float" with the data
as long as it remains in the ROS.
The ROS can be spanned by a parametric model
as long as (a)
for some parameter , and (b)
is a sufficient statistic for .
We can easily meet these conditions using
the multi-variate TED distribution (6.5) with
Condition (a) is met by
. Condition (b)
is met since (6.5) can be written
for some function . It therefore follows that
for any , where
is defined as the
distribution of when
Since the ratio (16.16) does not depend on
, it makes sense to place at the
can be easily evaluated,
and that is the point where both
have their maximum value, that is to say
at the maximum likelihood (ML) point
At this point, we can apply the central limit theorem
. The mean is given by
is expected value,
is the TED mean (6.3).
Note that under
From this, we can solve for the variance of ,
The covariance of is therefore
where is the diagonal matrix with elements (16.17).
Finally, then, we apply (16.16) at
, to get
is the Gaussian distribution
and covariance .
This approach can then be compared numerically with the
reciprocal of (2.12).