# Efficient estimator

Template:Broader 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.[1]

## Finite-sample efficiency

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[ T ] = θ), then the Cramér–Rao inequality states the variance of this estimator is bounded from below:

${\displaystyle \operatorname {Var} [\,T\,]\ \geq \ {\mathcal {I}}_{\theta }^{-1},}$

where ${\displaystyle \scriptstyle {\mathcal {I}}_{\theta }}$ 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 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.[2]

Historically, finite-sample efficiency was an early optimality criterion. 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.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} • This notion of efficiency is restricted to the class of unbiased estimators. Since there are no good theoretical reasons to require that estimators are unbiased, this restriction is inconvenient. In fact, if we use mean squared error as a selection criterion, many biased estimators will slightly outperform the “best” unbiased ones. For example, in multivariate statistics for dimension three or more, the mean-unbiased estimator, sample mean, is inadmissible: Regardless of the outcome, its performance is worse than for example the James–Stein estimator.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=

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