Lecture19-2010 - Sucient Statistics Charles B. Moss July...

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Sufficient Statistics Charles B. Moss July 23, 2010 I. Data Reduction A. The typical mode of operation in statistics is to use information from a sample X 1 , ··· X 2 to make inferences about an unknown parameter θ . 1. Put slightly diFerently, the researcher summarizes the infor- mation in the sample (or the sample values) with a statistic. 2. Thus, any statistic T ( X ) summarizes the data, or reduces the information in the sample to a single number. We use only the information in the statistic instead of the entire sample. 3. Put in a slightly more mathematical formulation, the statistic partitions the sample space into two sets a) De±ning the sample space for the statistic T = { t : t = T ( x ) ,x X } (1) b) Thus, a given value of a sample statistic T ( x ) implies that the sample comes from a space of sets A t such that t T , A t = { x : T ( x )= t } . c) The second possibility (that is ruled out by observing a sample statistic of T ( x ))is A C t = { x : T ( x ) 6 = t } . 4. Thus, instead of presenting the entire sample, we could report
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Lecture19-2010 - Sucient Statistics Charles B. Moss July...

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