In statistics, and more specifically in estimation theory, a minimum-variance unbiased estimator (MVUE or MVU estimator) is an unbiased estimator of parameters, whose variance is minimized for all values of the parameters. If an estimator is unbiased, then its mean squared error is equal to its variance, i.e., Statistics is a type of data analysis whose practice includes the planning, summarizing, and interpreting of observations of a system possibly followed by predicting or forecasting of future events based on a mathematical model of the system being observed. ... Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data. ... In statistics, a biased estimator is one that for some reason on average over- or underestimates what is being estimated. ... In statistics, an estimator is a function of the known data that is used to estimate an unknown parameter; an estimate is the result from the actual application of the function to a particular set of data. ... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ... In statistics the mean squared error of an estimator T of an unobservable parameter θ is i. ...

This follows immediately from the fact that the mean squared error is the sum of the variance and the square of the bias:

Consequently, if an estimator is unbiased, then minimizing its mean squared error is the same as minimizing its variance.

In many cases, a biased estimator can have a uniformly smaller mean squared error than does any unbiased estimator of the same parameter. See bias (statistics) for more. In statistics, a biased estimator is one that for some reason on average over- or underestimates what is being estimated. ...

If a MVUE is a complete statistic, then it is the only MVUE. In many cases, the Lehmann-Scheffé theorem can be used to show that an estimator is the unique MVUE. Constructing such an estimator is often done by relying on the Rao-Blackwell theorem. Suppose a random variable X (which may be a sequence (X1, ..., Xn) of scalar-valued random variables), has a probability distribution belonging to a known family of probability distributions, parametrized by θ, which may be either vector- or scalar-valued. ... In statistics, the Lehmann-Scheffé theorem states the any estimator that is complete, sufficient, and unbiased is the unique best unbiased estimator of its expectation. ... In statistics, the Rao-Blackwell theorem describes a technique that can transform an absurdly crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar criteria. ...