Slides19-2010 - Mean Squared Error and Maximum Likelihood...

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Mean Squared Error and Maximum Likelihood: Lecture XVIII Charles B. Moss October 18, 2010 Charles B. Moss () Mean Squared Error October 18, 2010 1 / 10
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1 Data Reduction 2 Sufficiency Principle Charles B. Moss () Mean Squared Error October 18, 2010 2 / 10
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Data Reduction 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 . Put slightly differently, the researcher summarizes the information in the sample (or the sample values) with a statistic. 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. Charles B. Moss () Mean Squared Error October 18, 2010 3 / 10
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Continued Put in a slightly more mathematical formulation, the statistic partitions the sample space into two sets Defining the sample space for the statistic T = { t : t = T ( x ) , x X } (1) 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 } .
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