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Unformatted text preview: Lecture 11 11.1 Sufficient statistic. (Textbook, Section 6.7) We consider an i.i.d. sample X 1 , . . . , X n with distribution θ from the family { θ : θ ∈ Θ } . Imagine that there are two people A and B, and that 1. A observes the entire sample X 1 , . . . , X n , 2. B observes only one number T = T ( X 1 , . . . , X n ) which is a function of the sample. Clearly, A has more information about the distribution of the data and, in par ticular, about the unknown parameter θ. However, in some cases, for some choices of function T (when T is so called sufficient statistics) B will have as much information about θ as A has. Definition. T = T ( X 1 , ··· , X n ) is called sufficient statistics if θ ( X 1 , . . . , X n  T ) = ( X 1 , . . . , X n  T ) , (11.1) i.e. the conditional distribution of the vector ( X 1 , . . . , X n ) given T does not depend on the parameter θ and is equal to . If this happens then we can say that T contains all information about the param eter θ of the disribution of the sample, since given T the distribution of the sample is always the same no matter what θ is. Another way to think about this is: why the second observer B has as much information about θ as observer A? Simply, given T , the second observer B can generate another sample X 1 , . . . , X, ....
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 Spring '09
 DmitryPanchenko
 Statistics, Conditional Probability, Probability theory, Xn, exponential family

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