Efficiency of estimators and UMVUE

Question

(1) An estimator is efficient when it reaches the Cramer Rao Lower Bound. (2) If an estimator reaches the CRLB, then it is the UMVUE. (3) The UMVUE is always unique.

If these points are correct, can we argue: (a) that if an efficient estimator exists, then it is always unique? (b) That an efficient estimator is always the UMVUE?

Moreover, if an estimator is not efficient (i.e. it does not reach the CRLB), can we say something about whether it is the UMVUE or not? Thanks!

Answer 1

You forgot about the "unbiasedness" restriction. Namely, your claims sounds ok as long as you add that the estimator is also unbiased. I.e., if an unbiased estimator reaches the CRLB for every $\theta$, then it is the UMVUE.

If these points are correct, can we argue: (a) that if an efficient estimator exists, then it is always unique? (b) That an efficient estimator is always the UMVUE?

• Only if it is unbiased. Otherwise - you can have an efficient biased estimator as well (that is clearly not a UMVUE).

Moreover, if an estimator is not efficient (i.e. it does not reach the CRLB), can we say something about whether it is the UMVUE or not?

• Generally no. UMVUE may not exist at all or could exist without reaching the CRLB. The CRLB is a theoretical bound, it does not imply that such an (unbiased) estimator, that reaches this bound, exists.