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Unformatted text preview: Suppose X ∼ P θ , θ ∈ Θ. Basu’s Lemma: If T ( X ) is complete and sufficient (for θ ∈ Θ), and S ( X ) is ancillary, then S ( X ) and T ( X ) are independent for all θ ∈ Θ. In other words, a complete sufficient statistic is independent of any ancillary statistic. Preliminary remarks (prior to proof): Let S = S ( X ), T = T ( X ). Let E θ denote expectation w.r.t. P θ . 1. The joint distribution of ( S,T ) depends on θ , so in general it is possible for S and T to be independent for some values of θ , but not for others. (Basu’s Lemma says this does not happen in this case.) 2. For any rv’s Y and Z , we know that E ( Y | Z ) = g ( Z ), i.e., the conditional expectation is a function of Z . If the joint distribution of ( Y,Z ) depends on a parameter θ , then E θ ( Y | Z ) = g ( Z,θ ), i.e., the conditional expectoration is a function of both Z and θ . (However, this function may turn out to be constant in one or both variables.) 3. In general, E ( Y ) = EE ( Y | Z ) and E θ ( Y ) = E θ E θ ( Y | Z ). 4. To show that Y and Z are independent, it suffices to show that L ( Y | Z ) = L ( Y ) which means that P ( Y ∈ A | Z ) = P ( Y ∈ A ) for all (Borel) sets A ....
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This note was uploaded on 12/15/2011 for the course STAT 5326 taught by Professor Frade during the Fall '10 term at FSU.
- Fall '10