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I have seen different formulations of the Lehmann Scheffe Theorem but am not sure if they are the same. For example, on wikipedia it states:

If $T$ is a complete sufficient statistic for $\theta$ and $E(g(T)) = \tau(\theta)$, then $g(T)$ is the uniformly minimum-variance unbiased estimator (UMVUE) of $\tau(\theta)$.

But here it states that:

If $T(X)$ is a complete sufficient statistic and $W(X)$ is an unbiased estimator of $τ(θ)$, then $φ(T) = E(W|T)$ is an UMVUE of $τ(θ)$. Furthermore, $φ(T)$ is the unique UMVUE in the sense that if $T^*$ is any other UMVUE, then $P\theta(φ(T) = T^*) = 1$ for all $θ$.

At a glance, it seems that the first wikipedia statement is needed for the second, and that the second uses the Rao-Blackwell Theorem. Are these two statements one and the same?

user321627
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1 Answers1

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In my inference notes I have the second statement as the theorem and the wikipedia statement as a corollary or application.

What's the difference?

Well, they are actually the same, maybe wikipedia statement is a particular case of the other, but there's no loss of generality. You are trying to get the UMVUE, and you know that it is $E[W|T]$ for a $W$ an unbiased estimator and $T$ sufficient and complete. But what Wikipedia says is that if you have the sufficient complete statistic then a second order function $h$ of $T$ will be an UMVUE for $E[h(T)]$ because $E[h(T)|T]=h(T)$. So if you have $h(T)$ unbiased you have directly the UMVUE, what it is more easy to calculate beacause you don't need conditioned distributions.

Skullgreymon
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