## 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 2017-05-19