I am trying to understand the difference between the UMVUE (uniformly minimum-variance unbiased estimator, also known as minimum-variance unbiased estimator (MVUE)) and the MVBE (minimum variance bound estimator).
There seems to be a lot of writing on the UMVUE, but not so much on the MVBE. What I have found that discusses this exact topic:
- This, which seems to indicate that a MVBE would also be a UMVUE (as the variance of a MVBE is smaller than the UMVUE).
- And this (see page 15), which also says that a MVBE is again the UMVUE.
However, I'm still unsure of the fundamental difference between the two.
- The MVBE is unbiased and attains (meaning it equals) the lower bound of the Cramer-Rao inequality (again from page 15 of that second source)
- "an unbiased estimator which achieves this [Cramer-Rao] lower bound is said to be (fully) efficient. Such a solution achieves the lowest possible mean squared error among all unbiased methods, and is therefore the [UMVUE]" (source).
Are these not both the same thing?