. . January 17th, 2013 Biostatistics 602 - Lecture 03 Hyun Min Kang January 17th, 2013 Hyun Min Kang Minimal Sufficient Statistics Lecture 03 Biostatistics 602 - Statistical Inference . Summary . . Minimal Sufficient Statistics Factorization . . . . . . . . 1 / 25 . . . . . . . . . . . . . . . . . . .
. . January 17th, 2013 Biostatistics 602 - Lecture 03 Hyun Min Kang dimension of the parameters? 4 Is the dimension of a sufficient statistic the always same to the . . Factorization Theorem? 3 What is an effective strategy to find sufficient statistics using the . . ? sufficient for 2 What is a necessary and sufficient condition for a statistic to be . . . . Last Lecture - Key Questions Summary . Minimal Sufficient Statistics Factorization . . . . . . . . 2 / 25 . . . . . . . . . . . . . . . . . . . 1 How do we show that a statistic is sufficient for θ ?
. . January 17th, 2013 Biostatistics 602 - Lecture 03 Hyun Min Kang dimension of the parameters? 4 Is the dimension of a sufficient statistic the always same to the . . Factorization Theorem? 3 What is an effective strategy to find sufficient statistics using the . . 2 What is a necessary and sufficient condition for a statistic to be . . . . Last Lecture - Key Questions Summary . Minimal Sufficient Statistics Factorization . . . . . . . . 2 / 25 . . . . . . . . . . . . . . . . . . . 1 How do we show that a statistic is sufficient for θ ? sufficient for θ ?
. . January 17th, 2013 Biostatistics 602 - Lecture 03 Hyun Min Kang dimension of the parameters? 4 Is the dimension of a sufficient statistic the always same to the . . Factorization Theorem? 3 What is an effective strategy to find sufficient statistics using the . . 2 What is a necessary and sufficient condition for a statistic to be . . . . Last Lecture - Key Questions Summary . Minimal Sufficient Statistics Factorization . . . . . . . . 2 / 25 . . . . . . . . . . . . . . . . . . . 1 How do we show that a statistic is sufficient for θ ? sufficient for θ ?
. . January 17th, 2013 Biostatistics 602 - Lecture 03 Hyun Min Kang dimension of the parameters? 4 Is the dimension of a sufficient statistic the always same to the . . Factorization Theorem? 3 What is an effective strategy to find sufficient statistics using the . . 2 What is a necessary and sufficient condition for a statistic to be . . . . Last Lecture - Key Questions Summary . Minimal Sufficient Statistics Factorization . . . . . . . . 2 / 25 . . . . . . . . . . . . . . . . . . . 1 How do we show that a statistic is sufficient for θ ? sufficient for θ ?
. . January 17th, 2013 Biostatistics 602 - Lecture 03 Hyun Min Kang . . Definition 6.2.1 . Recap - Sufficient Statistic Summary . Minimal Sufficient Statistics Factorization . . . . . . . . 3 / 25 . . . . . . . . . . . . . . . . . . . A statistic T ( X ) is a sufficient statistic for θ if the conditional distribution of sample X given the value of T ( X ) does not depend on θ .
. Summary January 17th, 2013 Biostatistics 602 - Lecture 03 Hyun Min Kang . . Theorem 6.2.2 . Recap - A Theorem for Sufficient Statistics . . Minimal Sufficient Statistics Factorization . . . . . . . . 4 / 25 . . . . . . . . . . . . . . . . . . . • Let f X ( x | θ ) is a joint pdf or pmf of X • and q ( t | θ ) is the pdf or pmf of T ( X ) . • Then T ( X ) is a sufficient statistic for θ , • if, for every x ∈ X , • the ratio f X ( x | θ )/ q ( T ( x ) | θ ) is constant as a function of θ .
. Summary January 17th, 2013 Biostatistics 602 - Lecture 03 Hyun Min Kang . . Theorem 6.2.6 - Factorization Theorem . Recap - Factorization Theorem . . Minimal Sufficient Statistics Factorization . . . . . . . . 5 / 25 . . . . . . . . . . . . . . . . . . . • Let f X ( x | θ ) denote the joint pdf or pmf of a sample X . • A statistic T ( X ) is a sufficient statistic for θ , if and only if • There exists function g ( t | θ ) and h ( x ) such that, • for all sample points x , • and for all parameter points θ , • f X ( x | θ ) = g ( T ( x ) | θ ) h ( x ) .
• An ordered statistic X X n X n are iid. • For any sufficient statistic T X , its one-to-one function q T X is also a sufficient statistic for . . Question . . . . . . . . Can we find a sufficient statistic that achieves the maximum data reduction? Hyun Min Kang Biostatistics 602 - Lecture 03 January 17th, 2013 . , if X . is always a sufficient statistic for . . . . . . . . Factorization Minimal Sufficient Statistics . Summary Minimal Sufficient Statistic . Sufficient statistics are not unique . . 6 / 25 . . . . . . . . . . . . . . . . . . . • T ( x ) = x : The random sample itself is a trivial sufficient statistic for any θ .
• For any sufficient statistic T X , its one-to-one function q T X . also a sufficient statistic for . . Question . . . . . . . . Can we find a sufficient statistic that achieves the maximum data reduction? Hyun Min Kang Biostatistics 602 - Lecture 03 January 17th, 2013 is 6 / 25 . Minimal Sufficient Statistics . . . . . . . . Factorization . Summary Minimal Sufficient Statistic . Sufficient statistics are not unique . . . . . . . . . . . . . . . . . . . . . • T ( x ) = x : The random sample itself is a trivial sufficient statistic for any θ . • An ordered statistic ( X (1) , · · · , X ( n ) ) is always a sufficient statistic for θ , if X 1 , · · · , X n are iid.
. . Question . . . . . . . . Can we find a sufficient statistic that achieves the maximum data reduction? Hyun Min Kang Biostatistics 602 - Lecture 03 January 17th, 2013 . 6 / 25 Summary . . . . . . Sufficient statistics are not unique . Minimal Sufficient Statistic . Minimal Sufficient Statistics Factorization . . . . . . . . . . . . . . . . . . . . . . . • T ( x ) = x : The random sample itself is a trivial sufficient statistic for any θ . • An ordered statistic ( X (1) , · · · , X ( n ) ) is always a sufficient statistic for θ , if X 1 , · · · , X n are iid. • For any sufficient statistic T ( X ) , its one-to-one function q ( T ( X )) is also a sufficient statistic for θ .
. Sufficient statistics are not unique January 17th, 2013 Biostatistics 602 - Lecture 03 Hyun Min Kang reduction? Can we find a sufficient statistic that achieves the maximum data . . Question . . . . . Minimal Sufficient Statistics . . . . . . . . Factorization . Summary Minimal Sufficient Statistic 6 / 25 . . . . . . . . . . . . . . . . . . . • T ( x ) = x : The random sample itself is a trivial sufficient statistic for any θ . • An ordered statistic ( X (1) , · · · , X ( n ) ) is always a sufficient statistic for θ , if X 1 , · · · , X n are iid. • For any sufficient statistic T ( X ) , its one-to-one function q ( T ( X )) is also a sufficient statistic for θ .
• The sample space • Given T X , can be partitioned into A t where • Maximum data reduction is achieved when • If size of . . consists of every possible sample - finest partition t t t T X for some x is minimal. . t t T x for some x is not less than , then can be called as a minimal sufficient statistic. Hyun Min Kang Biostatistics 602 - Lecture 03 January 17th, 2013 . . . Minimal Sufficient Statistic . . . . . . . . Factorization Minimal Sufficient Statistics . Summary . . Definition 6.2.11 . . . Why is this called ”minimal” sufficient statistic? . . . 7 / 25 . . . . . . . . . . . . . . . . . . . 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 ) .
• Given T X , can be partitioned into A t where • Maximum data reduction is achieved when • If size of t t t T X for some x is minimal. . . t T x for some x is not less than , then can be called as a minimal sufficient statistic. Hyun Min Kang Biostatistics 602 - Lecture 03 January 17th, 2013 t . . Why is this called ”minimal” sufficient statistic? . . . . . . . . Factorization Minimal Sufficient Statistics . Summary Minimal Sufficient Statistic . Definition 6.2.11 . . . 7 / 25 . . . . . . . . . . . . . . . . . . . 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
• Maximum data reduction is achieved when • If size of . t . . is minimal. t T x for some x . is not less than , then can be called as a minimal sufficient statistic. Hyun Min Kang Biostatistics 602 - Lecture 03 January 17th, 2013 Why is this called ”minimal” sufficient statistic? . 7 / 25 . . . . . . . . . Factorization Minimal Sufficient Statistics Summary Minimal Sufficient Statistic . Definition 6.2.11 . . . . . . . . . . . . . . . . . . . . . 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}
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