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.
Read more about this topic: Minimum-variance Unbiased Estimator
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