## ML approach

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

 (16.16)

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 point where can be easily evaluated, and that is the point where both and have their maximum value, that is to say at the maximum likelihood (ML) point

At this point, we can apply the central limit theorem to find . The mean is given by

where is expected value, and is the TED mean (6.3). Note that under ,

where [77,78]. From this, we can solve for the variance of ,

 (16.17)

The covariance of is therefore

where is the diagonal matrix with elements (16.17). Finally, then, we apply (16.16) at , and , to get

 (16.18)

where     is the Gaussian distribution with mean and covariance . This approach can then be compared numerically with the reciprocal of (2.12).

Baggenstoss 2017-05-19