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Are there efficient methods of showing when a UMVUE does not exist? I can think of the trivial case when no unbiased estimators exist at all. But that's not really interesting.

I feel like this would be difficult because there could be infinitely many unbiased estimators for a given problem. How would one efficiently compare the variances of each?

Thanks!

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  • $\begingroup$ This is not a kind of answer but might be a helpful short article, I want to share this : umass.edu/cluster/ed/unpublication/yr2000/c00ed78.PDF Good lucks~ $\endgroup$ – kurtkim Apr 25 '16 at 12:10
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    $\begingroup$ A useful theorem states that a necessary and sufficient condition for an unbiased estimator to be UMVUE is that it must be uncorrelated with every unbiased estimator of zero. This could be used to show that UMVUE of a certain parametric function does not exist. See here for example. $\endgroup$ – StubbornAtom Aug 26 '18 at 19:58

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