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Can efficient estimator be biased?
The fact that any efficient estimator is unbiased implies that the equality in (7.7) cannot be attained for any biased estimator. However, in all cases where an efficient estimator exists there exist biased estimators that are more accurate than the efficient one, possessing a smaller mean square error.
Why is efficient estimator important?
An efficient estimator is also the minimum variance unbiased estimator (MVUE). This is because an efficient estimator maintains equality on the Cramér–Rao inequality for all parameter values, which means it attains the minimum variance for all parameters (the definition of the MVUE).
How is an efficient estimator used in statistics?
In statistics, an efficient estimator is an estimator that estimates the quantity of interest in some “best possible” manner. The notion of “best possible” relies upon the choice of a particular loss function — the function which quantifies the relative degree of undesirability of estimation errors of different magnitudes.
What does relative efficiency of an estimator mean?
Equivalently, it’s the lower bound on the variance (the Cramer-Rao bound) divided by the variance of the estimator. The relative efficiency of two unbiased estimators is the ratio of their precisions (the bound cancelling out)
Are there any finite sample efficient estimators?
However this criterion has some limitations: Finite-sample efficient estimators are extremely rare. In fact, it was proved that efficient estimation is possible only in an exponential family, and only for the natural parameters of that family. This notion of efficiency is sometimes restricted to the class of unbiased estimators.
What does it mean by more efficient estimator cross validated?
It says in the above Wikipedia article that: If T 1 and T 2 are estimators for the parameter θ, then T 1 is said to dominate T 2 if: 1) If we don’t know θ , then how can we show one is smaller than the other in the above inequality.