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Rao-Blackwells theorem

  1. Apr 28, 2005 #1
    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.
  2. jcsd
  3. Apr 28, 2005 #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).
  4. Apr 29, 2005 #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:
    Last edited: Apr 29, 2005
  5. Apr 29, 2005 #4
    Yes, simplify the top expression, and it should become pretty clear. Your final answer should not surprise you.
  6. May 2, 2005 #5
    No, you're right. I got the arithmetic mean. Hopefully that's what you meant and I haven't done something very wrong.
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