On the existence of efficient estimators

The common signal processing problem of estimating some nonrandom parameters of a signal in additive noise is considered. The problem investigated in this paper is under what conditions an efficient estimator exists, i.e., an unbiased estimator with a variance equal to the Cramer-Rao lower bound (CRB). It is well known that if the signal is linear or, more generally, affine in the parameters and the noise Gaussian, an efficient estimator does exist. This paper shows that under some conditions, this is the only case where an efficient estimator exists.