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. Summary January 22th, 2013 Biostatistics 602 - Lecture 04 Hyun Min Kang January 22th, 2013 Hyun Min Kang Ancillary Statistics Lecture 04 Biostatistics 602 - Statistical Inference . . . . Location-scale Family Ancillary Statistics

  1. . Summary January 22th, 2013 Biostatistics 602 - Lecture 04 Hyun Min Kang January 22th, 2013 Hyun Min Kang Ancillary Statistics Lecture 04 Biostatistics 602 - Statistical Inference . . . . Location-scale Family Ancillary Statistics Minimal Sufficient Statistics . . . . . . . . . 1 / 23 . . . . . . . . . . . . . . . .

  2. . . January 22th, 2013 Biostatistics 602 - Lecture 04 Hyun Min Kang ? 5 How can we show that a statistic is minimal sufficient for . . 4 Is a minimal sufficient statistic unique? . 3 What is a minimal sufficient statistic? . . 2 What are examples obvious sufficient statistics for any distribution? . 1 Is a sufficient statistic unique? . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . Location-scale Family . Summary Recap from last lecture . 2 / 23 . . . . . . . . . . . . . . . .

  3. . . January 22th, 2013 Biostatistics 602 - Lecture 04 Hyun Min Kang ? 5 How can we show that a statistic is minimal sufficient for . . 4 Is a minimal sufficient statistic unique? . 3 What is a minimal sufficient statistic? . . 2 What are examples obvious sufficient statistics for any distribution? . 1 Is a sufficient statistic unique? . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . Location-scale Family . Summary Recap from last lecture . 2 / 23 . . . . . . . . . . . . . . . .

  4. . 4 Is a minimal sufficient statistic unique? 2 What are examples obvious sufficient statistics for any distribution? . . 3 What is a minimal sufficient statistic? . . . . . 5 How can we show that a statistic is minimal sufficient for ? Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 . 1 Is a sufficient statistic unique? . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . Location-scale Family . Summary Recap from last lecture . 2 / 23 . . . . . . . . . . . . . . . .

  5. . 4 Is a minimal sufficient statistic unique? 2 What are examples obvious sufficient statistics for any distribution? . . 3 What is a minimal sufficient statistic? . . . . . 5 How can we show that a statistic is minimal sufficient for ? Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 . 1 Is a sufficient statistic unique? . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . Location-scale Family . Summary Recap from last lecture . 2 / 23 . . . . . . . . . . . . . . . .

  6. . . January 22th, 2013 Biostatistics 602 - Lecture 04 Hyun Min Kang . . 4 Is a minimal sufficient statistic unique? . . 3 What is a minimal sufficient statistic? . . 2 What are examples obvious sufficient statistics for any distribution? . . 1 Is a sufficient statistic unique? . Minimal Sufficient Statistics . . . . . . . . . 2 / 23 . Ancillary Statistics Location-scale Family . Summary Recap from last lecture . . . . . . . . . . . . . . . . 5 How can we show that a statistic is minimal sufficient for θ ?

  7. . Minimal Sufficient Statistic January 22th, 2013 Biostatistics 602 - Lecture 04 Hyun Min Kang . . Why is this called ”minimal” sufficient statistic? . . . Definition 6.2.11 . . Summary Minimal Sufficient Statistics . . . . . . . . . . Ancillary Statistics Location-scale Family 3 / 23 . . . . . . . . . . . . . . . . A sufficient statistic T ( X ) is called a minimal sufficient statistic if, for any other sufficient statistic T ′ ( X ) , T ( X ) is a function of T ′ ( X ) . • The sample space X consists of every possible sample - finest partition • Given T ( X ) , X can be partitioned into A t where t ∈ T = { t : t = T ( X ) for some x ∈ X} • Maximum data reduction is achieved when |T | is minimal. • If size of T ′ = t : t = T ′ ( x ) for some x ∈ X is not less than |T | , then |T | can be called as a minimal sufficient statistic.

  8. . Theorem for Minimal Sufficient Statistics January 22th, 2013 Biostatistics 602 - Lecture 04 Hyun Min Kang . . In other words.. . . . Theorem 6.2.13 . . Summary . . . . . . . . . . Minimal Sufficient Statistics 4 / 23 Location-scale Family Ancillary Statistics . . . . . . . . . . . . . . . . • f X ( x ) be pmf or pdf of a sample X . • Suppose that there exists a function T ( x ) such that, • For every two sample points x and y , • The ratio f X ( x | θ )/ f X ( y | θ ) is constant as a function of θ if and only if T ( x ) = T ( y ) . • Then T ( X ) is a minimal sufficient statistic for θ . • f X ( x | θ )/ f X ( y | θ ) is constant as a function of θ = ⇒ T ( x ) = T ( y ) . • T ( x ) = T ( y ) = ⇒ f X ( x | θ )/ f X ( y | θ ) is constant as a function of θ

  9. f X x x i exp n . n . n i exp x i exp x i exp n i x i i . exp x i exp n i n i exp x i Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 . . . . . . . . . . . . . Minimal Sufficient Statistics Ancillary Statistics Location-scale Family . Summary Exercise from the textbook . Problem . . . Solution . . . 5 / 23 . . . . . . . . . . . . . . . . X 1 , · · · , X n are iid samples from e − ( x − θ ) f X ( x | θ ) = (1 + e − ( x − θ ) ) 2 , −∞ < x < ∞ , −∞ < θ < ∞ Find a minimal sufficient statistic for θ .

  10. x i exp n . x i . n exp n i x i n i exp exp Solution n i n i exp x i Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 . . . Problem . . . . . . . . . Minimal Sufficient Statistics Ancillary Statistics Location-scale Family . Summary Exercise from the textbook . . . 5 / 23 . . . . . . . . . . . . . . . . X 1 , · · · , X n are iid samples from e − ( x − θ ) f X ( x | θ ) = (1 + e − ( x − θ ) ) 2 , −∞ < x < ∞ , −∞ < θ < ∞ Find a minimal sufficient statistic for θ . exp ( − ( x i − θ )) ∏ f X ( x | θ ) = (1 + exp ( − ( x i − θ ))) 2 i =1

  11. x i exp n . . . n exp n i n . i exp x i Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 . Solution . Summary . . . . . . . . . Minimal Sufficient Statistics Ancillary Statistics Location-scale Family . 5 / 23 . Exercise from the textbook Problem . . . . . . . . . . . . . . . . . X 1 , · · · , X n are iid samples from e − ( x − θ ) f X ( x | θ ) = (1 + e − ( x − θ ) ) 2 , −∞ < x < ∞ , −∞ < θ < ∞ Find a minimal sufficient statistic for θ . exp ( − ∑ n exp ( − ( x i − θ )) i =1 ( x i − θ )) ∏ f X ( x | θ ) = (1 + exp ( − ( x i − θ ))) 2 = ∏ n i =1 (1 + exp ( − ( x i − θ ))) 2 i =1

  12. . . January 22th, 2013 Biostatistics 602 - Lecture 04 Hyun Min Kang n . . Solution . . . Problem . Exercise from the textbook Ancillary Statistics . . . Summary . . . . . . Minimal Sufficient Statistics 5 / 23 . Location-scale Family . . . . . . . . . . . . . . . . X 1 , · · · , X n are iid samples from e − ( x − θ ) f X ( x | θ ) = (1 + e − ( x − θ ) ) 2 , −∞ < x < ∞ , −∞ < θ < ∞ Find a minimal sufficient statistic for θ . exp ( − ∑ n exp ( − ( x i − θ )) i =1 ( x i − θ )) ∏ f X ( x | θ ) = (1 + exp ( − ( x i − θ ))) 2 = ∏ n i =1 (1 + exp ( − ( x i − θ ))) 2 i =1 exp ( − ∑ n i =1 x i ) exp ( n θ ) = i =1 (1 + exp ( − ( x i − θ ))) 2 ∏ n

  13. x n are permutations X n . i x i n i exp y i exp n i y i n x i exp n The ratio above is constant to if and only if x of y y n . So the order statistic T X X is a minimal sufficient statistic. Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 i exp . . . . . . . . . . . Minimal Sufficient Statistics Ancillary Statistics Location-scale Family . Summary Solution (cont’d) Applying Theorem 6.2.13 . . 6 / 23 . . . . . . . . . . . . . . . . i =1 (1 + exp ( − ( y i − θ ))) 2 f X ( x | θ ) exp ( − ∑ n i =1 x i ) exp ( n θ ) ∏ n = i =1 (1 + exp ( − ( x i − θ ))) 2 f X ( y | θ ) exp ( − ∑ n i =1 y i ) exp ( n θ ) ∏ n

  14. x n are permutations X n . . January 22th, 2013 Biostatistics 602 - Lecture 04 Hyun Min Kang sufficient statistic. is a minimal X y n . So the order statistic T X of y if and only if x The ratio above is constant to . . 6 / 23 Applying Theorem 6.2.13 . . . . . . . . . . Solution (cont’d) Minimal Sufficient Statistics Ancillary Statistics Location-scale Family . Summary . . . . . . . . . . . . . . . . i =1 (1 + exp ( − ( y i − θ ))) 2 f X ( x | θ ) exp ( − ∑ n i =1 x i ) exp ( n θ ) ∏ n = i =1 (1 + exp ( − ( x i − θ ))) 2 f X ( y | θ ) exp ( − ∑ n i =1 y i ) exp ( n θ ) ∏ n i =1 (1 + exp ( − ( y i − θ ))) 2 exp ( − ∑ n i =1 x i ) ∏ n = exp ( − ∑ n i =1 y i ) ∏ n i =1 (1 + exp ( − ( x i − θ ))) 2

  15. . Summary January 22th, 2013 Biostatistics 602 - Lecture 04 Hyun Min Kang sufficient statistic. . . . Applying Theorem 6.2.13 . Solution (cont’d) 6 / 23 . . Ancillary Statistics Minimal Sufficient Statistics . . . . . . . . Location-scale Family . . . . . . . . . . . . . . . . i =1 (1 + exp ( − ( y i − θ ))) 2 f X ( x | θ ) exp ( − ∑ n i =1 x i ) exp ( n θ ) ∏ n = i =1 (1 + exp ( − ( x i − θ ))) 2 f X ( y | θ ) exp ( − ∑ n i =1 y i ) exp ( n θ ) ∏ n i =1 (1 + exp ( − ( y i − θ ))) 2 exp ( − ∑ n i =1 x i ) ∏ n = exp ( − ∑ n i =1 y i ) ∏ n i =1 (1 + exp ( − ( x i − θ ))) 2 The ratio above is constant to θ if and only if x 1 , · · · , x n are permutations of y 1 , · · · , y n . So the order statistic T ( X ) = ( X (1) , · · · , X ( n ) ) is a minimal

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