I have some problems in understanding the concept of efficiency as related to an estimator. My sources (Mukhopadhyay, 2000 and Casella, Berger, 2002) do not treat this argument as I expected since they analyse only the concept of asymptotic efficiency.
I do not understand if there exist a concept of efficiency valid also (or only) in finite samples. Or if efficiency is not a per se concept and it is used only to compare estimators, talking about more efficient estimator and so on.
What I know is that an estimator is efficient if it reaches the Cramér-Rao Lower Bound. But this is a characterization, it is not a definition. And, moreover, Cramér-Rao inequality refers only to a subset of estimators, those which are unbiased for a certain $\tau(\theta)$. The concept of efficiency is meaningful only in the case of unbiased estimators?
If someone could provide me some sources or a brief excursus about the concept of efficiency and efficient estimators, I will be grateful.