Lecture 19-2007 - Sufficient Statistics Lecture XIX I. Data...

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1 Sufficient Statistics Lecture XIX I. Data Reduction A. References: Casella, G. and R.L. Berger Statistical Inference 2 nd Edition, New York: Duxbury Press, Chapter 6 “Principles of Data Reduction.” Pp 271-309. Hogg, R.V., A. Craig, and J.W. McKean Introduction to Mathematical Statistics 6 th Edition, Englewood Cliffs, New Jersey: Prentice Hall, 2004, Chapter 7 “Sufficiency,” pp 367-418. B. The typical mode of operation in statistics is to use information from a sample 1 , n XX to make inferences about an unknown parameter . 1. Put slightly differently, the researcher summarizes the information in the sample (or the sample values) with a statistic. 2. Thus, any statistic   TX 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) Defining the sample space for the statistic     :, t t T x x     b) Thus, a given value of a sample statistic   Tx implies that the sample comes from a space of sample sets t A such that t  ,     : t A x T x t  . c) The second possibility (that is ruled out by observing a sample statistic of   ) is     : C t A x T x t  .
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Lecture 19-2007 - Sufficient Statistics Lecture XIX I. Data...

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