03_Estimation_part4

03_Estimation_part4 - Ch8 p.53 Data reduction the concepts...

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NTHU MATH 2820, 2008, Lecture Notes Ch8, p.53 Data reduction --- the concepts of sufficiency, minimal sufficiency, and completeness Question 6.4 (information and data reduction) raw (original) data order statistics histogram sample mean sample variance To present concrete information, the data reduction through useful transfor- mations is required. However, non-invertible transformations can cause the loss of information. ( Example? ) The lost information can be important or worthless to the objective of studying the data. How to examine whether the important information lose in transformation? Furthermore, what is impor- tant information ? 1 n n i =1 ( X i X n ) 2 X 1 , X 2 , . . . , X n X n X (1) , X (2) , . . . , X ( n ) (numerical or graphical) transformations of data appear all the time in statis- tics for o ff ering a summary of information contained in data. For example, made by Shao-Wei Cheng (NTHU, Taiwan) Ch8, p.54 Summary (formulation of information and data reduction problem, TBp. 305) Let X 1 , X 2 , . . . , X n be a sample with joint pdf/pmf f ( x | Θ ), wherer Θ is unknown parameter. X 1 , X 2 , . . . , X n contains two types of information: information related to Θ information irrelevant to Θ For example, toss a coin n times, i.e., X 1 , X 2 , . . ., X n are i.i.d. from Bernoulli B ( θ ), X n or T = n i =1 X i contains information about θ When T is known, the information that at which trials the head’s occur is irrelevant to θ n =5, consider the following possible results: B (0, 1, 1, 1, 1), T = 4; (1, 0, 1, 1, 1), T = 4; (1, 1, 0, 1, 1), T = 4; (1, 1, 1, 0, 1), T = 4; (1, 1, 1, 1, 0), T = 4 B (1, 0, 0, 0, 0), T = 1; (0, 1, 0, 0, 0), T = 1; (0, 0, 1, 0, 0), T = 1; (0, 0, 0, 1, 0), T = 1; (0, 0, 0, 0, 1), T = 1
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NTHU MATH 2820, 2008, Lecture Notes Ch8, p.55 Definition 6.15 (sufficient, TBp. 305)
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