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Math Help - Statistics: Rao-Blackwell Theorem

  1. #1
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    Statistics: Rao-Blackwell Theorem



    I was reading this theorem again and again, but I really don't understand the meaning of "for all theta" in this theorem (circled in red). I thought that theta is supposed to be an unkown CONSTANT. Then how would it make sense to say "for all theta"? How would it change the theorem if the "for all theta" wasn't there? Why is this "for all theta" absolutely necessary?
    [If it were to say "for all theta hat", then it would at least make some sense to me since theta hat is a random variable...at least it's a variable and it makes sense to talk about "for all", not a constant like theta]

    Could someone please explain? Thank you for any help!
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  2. #2
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    Let's consider the following example:

    1) To apply the thoerem, we first need a sufficient statistic. This is easy by the factorization theorem. Next, we need unbiased estimator for theta which is harder to find. Is there a general way to pinpoint the correct one?
    In the solutions they found an unbiased estimator by first computing E(Yi^2). It looks like there is some "GUESSING" factor in here. How can we think of computing E(Yi^2) in the first place? There are many possible functions of the sufficient statistic. Why choose the particular one that they've chosen? Where to get this inspiration? Is there a systmatic way to find such unbiased esimtators?

    2) What I understand about the theorem now is that it does NOT give the esimator with the smallEST variance (only smallER), but in the above example, they immediately concluded from the theorem that it is the MINIMUM variance unbiased estimator. How come?

    Thanks for explaining!
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