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Unformatted text preview: Sufficient Statistics Data Reduction We want to summarize the information in the sample by determining a few key features of the sample values. Done by computing statistics. We want to keep all the information that we need, but discard everything we do not need. Definition (Sufficient statistic) X 1 , , X n is a random sample from f ( x ; ), . Y 1 = u 1 ( X 1 , , X n ) is a statistic with pdf g 1 ( y 1 ; ). Then Y 1 = u 1 ( X 1 , , X n ) is a sufficient statistic for a parameter if and only if f ( x 1 ; ) f ( x n ; ) g 1 ( u 1 ( x 1 , , x n ); ) = H ( x 1 , , x n ) where H ( x 1 , , x n ) does not depend on . 1 In more general situations, X 1 , , X n do not need to be independent or identically distributed. In this case, Y 1 is a sufficient statistic for iff f ( x 1 , x 2 , , x n ; ) g 1 ( u 1 ( x 1 , , x n ); ) = H ( x 1 , , x n ) does not depend on ....
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This note was uploaded on 09/10/2009 for the course STATS 517 taught by Professor Song during the Fall '07 term at Purdue University-West Lafayette.
- Fall '07