Cramer-Rao lower bound and efficiency vs biased estimator efficiency I am a bit confused on the Cramer-Rao (CR) lower bound.
If an estimator achieves the CR lower bound, then it is UMVUE, right?
And for any given set of unbiased estimators, the one with the lowest variance is the most efficient.
So for any set of unbiased estimators, the one that achieves the CR lower bound is the most efficient of the group since it is uniformly min-var., but is it possible to find a biased estimator that could be more efficient?
If I did find one, is comparing the two really worth while, or should I just stick with the CR lower bound estimator. 
Thanks for looking
 A: Yes. Think of any biased but constant estimator, of which variance is zero. Then, it is POSSIBLE to find such an estimator.
Also, refer to "Petre Stoica, Randolph L Moses. On biased estimators and the unbiased Cramér-Rao lower bound, 1990"
A: An efficient estimator is an estimator that achieves the CR lower bound.
An efficient estimator has lowest mean square error among all unbiased estimators.
Note that all these refer to the fact that the considered class of estimators is unbiased.
Furthermore, there is no ordering in efficiency. An estimator either is efficient (it is unbiased and achieves the CR), or it is not efficient. The statement "more efficient" has no statistical meaning, so you shoukd consider a risk measure such as MSE.
Indeed, there are many biased estimators that have lower MSE. For instance the James-Stein estimator is a biased estimator that achieves uniformly lower MSE that the least squares estimator (the UMVUE) for the estimation of the mean of a Gaussian random variable. Other example would be Bayesian linear regression and Tikhonov regularization.
