Rao-Blackwells Theorem: Efficient Estimation Using Sufficient Statistics

[Zaare]

Given the facts

1. $$X_1 ,...,X_n$$ are independent and have the same distribution.

2. The expectation value of $$X_i$$ is $$E\left( {X_i } \right) = \theta$$.

3. $$T=\sum\limits_{i = 1}^n {X_i }$$ is a sufficient statistic.

I'm asked to find an astimate for $$\theta$$ starting with the estimate $$U=X_1$$.

According to Rao-Blackwells theorem, the new estimate is taken as $$g(t)=E(U|T=t)$$.

I don't know how to calculate this expression further. Any help or tip would be appreciated.

 

[juvenal]

I would calculate the sum of the expectation value of X_i conditioned on the sufficient statistic. That sum can then be equated to n*g(t).

 

[Zaare]

Ok, I think I get it. You mean I should calculate this:
$$\sum\limits_{i = 1}^n {E\left( {X_i |T = t} \right)}$$

And that would equal this:
$$ng(t)=nE(U|T=t)=nE(X_1|T=t)$$

 

[juvenal]

Yes, simplify the top expression, and it should become pretty clear. Your final answer should not surprise you.

 

[Zaare]

No, you're right. I got the arithmetic mean. Hopefully that's what you meant and I haven't done something very wrong.