UMS
We now consider UMS  that is
to generate samples of data which will have
exactly the specified ML parameter estimates,
uniformly on the manifold.
Given fixed MLE values of
,
,
the we can easily define the manifold of all that lead to the
given MLE as follows. Define the statistics
,
,
and
.
We can use (4.2), and (4.3) to compute
.
The ML estimates
are such that the partial derivative of
with respect
to each parameter is zero. The derivative constraint for
leads to

(4.7) 
We can use (4.7) to compute .
So, we are able to compute
just from
.
The equations defining
lead
to a set of constraints for that can be written in the form
where
This is the problem of Section 4.4. We can apply the results
of that Section to sample the manifold. Every sample will meet the derivative constraint
for
as well as produce the same amplitude and variance estimates,
so will produce the given ML solution. See
software/test_ml.m for an example
of sinusoidal frequency estimation.
See also Section 8.2.3 for an example of this method.
Baggenstoss
20170519