I'm reading proof of Lehmann-Scheffe theorem in Casella's Statistical Inference (2nd ed)
Theorem states that
Let $T$ be a complete sufficient statistic for a parameter $\theta$, and let $\phi(T)$be any estimator based only on T.
Then $\phi(T)$is the unique best unbiased estimator of its expected value
To prove $\phi(T)$ is best unbiased estimator, he use following lemma
Suppose $EW = \tau(\theta)$. Then $W$ is best unbiased estimator of $\tau(\theta)$ if and only if $W$ is uncorrelated with all unbiased estimator of $0$
Then he states proof of Lehmann-Scheffe theorem as follows:
- Since $T$ is complete, there are no unbiased estimator of $0$ that are based on $T$
- By Rao-Blackwell Theorem, It suffices to consider unbiased estimator of $0$ based on $T$
- Hence, $\phi(T)$ is uncorrelated to all unbiased estimator of $0$ (as there is only one unbiased estimator of $0$, just $0$ itself.)
- So by lemma, we got desired result
I cannot understand second step.
If we deal with to find or check best unbiased estimator, I think we can use Rao-Blackwell theorem since estimator based on $T$ always has smaller variance.
But in here, we want to check whether $\phi(T)$ is uncorrelated to all unbiased estimator of $0$, so I think it's not sufficient to check only estimator based on $T$.
How can I solve this problem?
(Indeed, I have other proof of Lehmann-Scheffe theorem. I just want to know how this proof make sense)
Consider the possibility that there is a better unbiased estimator. The Rao-Backwell theorem says that the expected value of that unbiased estimator given $T$ has variance that is less than or equal to the original estimator. That means this improved estimator (which is a function of $T$) can be considered instead as the "better estimator". I other words, if I can show my estimator beats that one, then it also beats the original one.
Can you tell me where this proof is? The statement is Theorem 7.3.23, but I don't see the proof.
Your question is related to this one.