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In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.
MINQUE estimators also provide an alternative to maximum likelihood estimators or restricted maximum likelihood estimators for variance components in mixed effects models. [3] MINQUE estimators are quadratic forms of the response variable and are used to estimate a linear function of the variances.
This may occur either if for any unbiased estimator, there exists another with a strictly smaller variance, or if an MVU estimator exists, but its variance is strictly greater than the inverse of the Fisher information. The Cramér–Rao bound can also be used to bound the variance of biased estimators of given bias.
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. [6] Historically, finite-sample efficiency was an early optimality criterion. However this criterion has some limitations:
Among unbiased estimators, there often exists one with the lowest variance, called the minimum variance unbiased estimator . In some cases an unbiased efficient estimator exists, which, in addition to having the lowest variance among unbiased estimators, satisfies the Cramér–Rao bound , which is an absolute lower bound on variance for ...
So δ 1 is clearly a very much improved estimator of that last quantity. In fact, since S n is complete and δ 0 is unbiased, δ 1 is the unique minimum variance unbiased estimator by the Lehmann–Scheffé theorem.
In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) [1] states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in the linear regression model are uncorrelated, have equal variances and expectation value of zero. [2]
and therefore the function is the unique function of Y with variance not greater than that of any other unbiased estimator. We conclude that φ ( Y ) {\displaystyle \varphi (Y)} is the MVUE. Example for when using a non-complete minimal sufficient statistic