Improvements on the Cramer-Rao bound

In the context of nonrandom parameter estimation from a finite set of observations, the situation associated with weak Cramer-Rao bounds is addressed. It is shown that it is possible to improve the Cramer-Rao bound by incorporating higher-order derivatives of the joint probability density function (PDF) such that the sequence (of bounds) so generated converges to a definite limit. The limit so obtained is shown to coincide with the variance of the actual uniformly minimum variance unbiased estimator (UMVUE) for this parameter, provided the underlying PDF is of the Korpman-Darmois exponential family. 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 useful, since this procedure only involves the first-/and higher-order derivatives of the joint PDF and their cross covariance