Rao-Blackwells Theorem: Efficient Estimation Using Sufficient Statistics

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In summary, given the facts that X_1,...,X_n are independent and have the same distribution, with an expectation value of E(X_i)=\theta and a sufficient statistic T=\sum\limits_{i=1}^n X_i, we can use Rao-Blackwells theorem to calculate the new estimate for \theta as g(t)=E(U|T=t) by finding the sum of the expectation values of X_i conditioned on the sufficient statistic and equating it to n*g(t). This results in the arithmetic mean, which is the final answer.
  • #1
Zaare
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Given the facts

1. [tex] X_1 ,...,X_n [/tex] are independent and have the same distribution.

2. The expectation value of [tex]X_i[/tex] is [tex] E\left( {X_i } \right) = \theta [/tex].

3. [tex]T=\sum\limits_{i = 1}^n {X_i }[/tex] is a sufficient statistic.

I'm asked to find an astimate for [tex]\theta[/tex] starting with the estimate [tex]U=X_1[/tex].

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

I don't know how to calculate this expression further. Any help or tip would be appreciated.
 
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  • #2
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).
 
  • #3
Ok, I think I get it. You mean I should calculate this:
[tex]\sum\limits_{i = 1}^n {E\left( {X_i |T = t} \right)} [/tex]

And that would equal this:
[tex]ng(t)=nE(U|T=t)=nE(X_1|T=t)[/tex]
 
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  • #4
Yes, simplify the top expression, and it should become pretty clear. Your final answer should not surprise you.
 
  • #5
No, you're right. I got the arithmetic mean. Hopefully that's what you meant and I haven't done something very wrong.
 

FAQ: Rao-Blackwells Theorem: Efficient Estimation Using Sufficient Statistics

What is the Rao-Blackwells theorem?

The Rao-Blackwells theorem is a fundamental concept in statistics that states that the conditional expectation of a statistic given a sufficient statistic is equal to the statistic itself. In other words, the best unbiased estimator of a parameter can be found by conditioning on a sufficient statistic.

Who developed the Rao-Blackwells theorem?

The Rao-Blackwells theorem was developed by Indian mathematician Calyampudi Radhakrishna Rao and American statistician David Blackwell in the 1940s.

What is the importance of the Rao-Blackwells theorem?

The Rao-Blackwells theorem is important because it allows for the improvement of statistical estimators by using sufficient statistics. This can lead to more efficient and accurate estimations of parameters in various statistical models.

How is the Rao-Blackwells theorem used in practice?

In practice, the Rao-Blackwells theorem is often used in the development of statistical methods and models. It can also be used to improve the efficiency of estimators in various fields such as economics, engineering, and biology.

What are some limitations of the Rao-Blackwells theorem?

While the Rao-Blackwells theorem is a powerful tool in statistics, it does have some limitations. It only applies to unbiased estimators and requires the existence of a sufficient statistic. Additionally, it may not always result in the most efficient estimator.

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