Higher order Cramer-Rao bounds and UMVUEs

This work addresses the situation associated with weak Cramer-Rao (C-R) bounds and shows that it is possible to improve the C-R bound by incorporating higher order derivatives of the joint pdf such that the sequence (of bounds) so generated converges to a definite limit. Further, the limit so obtained is shown to coincide with the variance of the actual uniformly minimum variance unbiased estimator (UMVUE) for that parameter, provided the underlying pdf is of the Korpman-Darmois exponential family. Interestingly, under these conditions, this technique also provides a constructive procedure to evaluate the UMVUE for any unknown parameter (or its functions) contained in the data. From a practical standpoint this is extremely useful, since this procedure only involves first and higher order derivatives of the joint pdf and their cross covariances.