The theorem states that any estimator which is unbiased for a given unknown quantity and that depends on the data only through a complete, sufficient statistic is the unique best unbiased estimator of that quantity.

Is UMVUE unique?

1 Answer. Generally, an UMVUE is essentially unique. The estimator you provided is not an UMVUE though, indeed it is not even unbiased!! Notice that E[1−X]=1−E[X]=1−p provided that our random variable is a Bernoulli with parameter p.

Does unbiased estimator always exist?

It is important to note that a uniformly minimum variance unbiased estimator may not always exist, and even if it does, we may not be able to find it. There is not a single method that will always produce the MVUE. One useful approach to finding the MVUE begins by finding a sufficient statistic for the parameter.