# Minimum-variance unbiased estimator

In statistics a uniformly minimum-variance unbiased estimator or minimum-variance unbiased estimator (UMVUE or MVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For practical statistics problems, it is important to determine the UMVUE if one exists, since less-than-optimal procedures would naturally be avoided, other things being equal. This has led to substantial development of statistical theory related to the problem of optimal estimation. While the particular specification of "optimal" here requiring unbiasedness and measuring "goodness" using the variance may not always be what is wanted for any given practical situation, it is one where useful and generally applicable results can be found.

## Definition

Consider estimation of based on data i.i.d. from some member of a family of densities , where is the parameter space. An unbiased estimator of is UMVUE if ,

for any other unbiased estimator

If an unbiased estimator of exists, then one can prove there is an essentially unique MVUE. Using the Rao–Blackwell theorem one can also prove that determining the MVUE is simply a matter of finding a complete sufficient statistic for the family and conditioning any unbiased estimator on it.

Further, by the Lehmann–Scheffé theorem, an unbiased estimator that is a function of a complete, sufficient statistic is the UMVUE estimator.

Put formally, suppose is unbiased for , and that is a complete sufficient statistic for the family of densities. Then

is the MVUE for

A Bayesian analog is a Bayes estimator, particularly with minimum mean square error (MMSE).

## Estimator selection

An efficient estimator need not exist, but if it does and if it is unbiased, it is the MVUE. Since the mean squared error (MSE) of an estimator δ is

the MVUE minimizes MSE among unbiased estimators. In some cases biased estimators have lower MSE because they have a smaller variance than does any unbiased estimator; see estimator bias.

## Example

Consider the data to be a single observation from an absolutely continuous distribution on with density

and we wish to find the UMVU estimator of

First we recognize that the density can be written as

Which is an exponential family with sufficient statistic . In fact this is a full rank exponential family, and therefore is complete sufficient. See exponential family for a derivation which shows

Therefore

Clearly is unbiased, thus the UMVU estimator is

This example illustrates that an unbiased function of the complete sufficient statistic will be UMVU, as Lehmann–Scheffé theorem states.

## Other examples

where m is the sample maximum. This is a scaled and shifted (so unbiased) transform of the sample maximum, which is a sufficient and complete statistic. See German tank problem for details.

## References

• Keener, Robert W. (2006). Statistical Theory: Notes for a Course in Theoretical Statistics. Springer. pp. 47–48, 57–58.
• Voinov V. G.,, Nikulin M.S. (1993). Unbiased estimators and their applications, Vol.1: Univariate case. Kluwer Academic Publishers. pp. 521p.
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