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Unformatted text preview: M4056 Lecture Notes. Wednesday, September 22, 2010 Theorem 6.2.6. T ( vector X ) is a sufficient statistic for if and only if there are functions g ( t | ) and h ( vectorx ) such that: f vector X ( vectorx | ) = g ( T ( vectorx ) | ) h ( vectorx ) . Remark. We will present the proof for the discrete case. The continuous case requires some additional assumptions (which are usually met in practice) and the argument requires attention to some details that do not arise in the discrete case. The central ideas are the same, though. Proof. Note that p vector X ( vectorx | ) = P ( vector X = vectorx ) . ( ** ) Also, using conditional probability as in ( * ) in the last lecture: P ( vector X = vectorx ) = P ( T ( vector X ) = T ( vectorx )) P ( vector X = vectorx | T ( vector X ) = T ( vectorx )) . ( * * * ) ( ) Suppose T is sufficient for . We need to show that P ( vector X = vectorx ) factors in the stated way. Let g ( t | ) := P ( T ( vector X...
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This note was uploaded on 11/29/2011 for the course MATH 4056 taught by Professor Staff during the Fall '08 term at LSU.
- Fall '08