How to prove the UMVUE of $ET^k$ is $T^k$?

Asked Sep 24 '24 at 06:10

Active Sep 24 '24 at 07:01

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Here is the problem. Suppose $T$ is the UMVUE of $\theta$, please prove the UMVUE of $ET^k$ is $T^k$(k=1,2,3,...), where $ET^{2k}<\infty$ holds.

I have a fake proof by using L-S theorem: Because T is the UMVUE of $\theta$, T should be sufficient and complete, (which is not true). Then use L-S theorem, it proved.

But I don't know how to proof it correctly. Please help me.

edited Sep 24 '24 at 07:01

asked Sep 24 '24 at 06:10

UMVUE

Did you accidentially reverse $T^k$ and $ET^k$ in the question? How is $k$ defined? – Zack Fisher Sep 24 '24 at 06:46
Oops, my bad. I've edited it. – UMVUE Sep 24 '24 at 07:02

How to prove the UMVUE of $ET^k$ is $T^k$?

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