Efficient estimator

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. The most common choice of the loss function is quadratic, resulting in the mean squared error criterion of optimality. 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. The most common choice of the loss function is quadratic, resulting in the mean squared error criterion of optimality. Suppose { Pθ | θ ∈ Θ } is a parametric model and X = (X1, …, Xn) are the data sampled from this model. Let T = T(X) be an estimator for the parameter θ. If this estimator is unbiased (that is, E = θ), then the Cramér–Rao inequality states the variance of this estimator is bounded from below: where I θ {displaystyle scriptstyle {mathcal {I}}_{ heta }} is the Fisher information matrix of the model at point θ. Generally, the variance measures the degree of dispersion of a random variable around its mean. Thus estimators with small variances are more concentrated, they estimate the parameters more precisely. We say that the estimator is a finite-sample efficient estimator (in the class of unbiased estimators) if it reaches the lower bound in the Cramér–Rao inequality above, for all θ ∈ Θ. Efficient estimators are always minimum variance unbiased estimators. However the converse is false: There exist point-estimation problems for which the minimum-variance mean-unbiased estimator is inefficient.