Data Reduction - Summary Lecture 08 Biostatistics 602 - Statistical - PowerPoint PPT Presentation

. . February 5th, 2013 Biostatistics 602 - Lecture 08 Hyun Min Kang February 5th, 2013 Hyun Min Kang Data Reduction - Summary Lecture 08 Biostatistics 602 - Statistical Inference . Summary . . Review Exponential Family . . . . . . . . . . . . . . . . 1 / 24 . . . . . . . . . .

. 2 Does a Bernoulli distribution belongs to an exponential family? February 5th, 2013 Biostatistics 602 - Lecture 08 Hyun Min Kang online. Visit http://www.polleverywhere.com/survey/laGysmUTS to respond 5 When can the sufficient statistic be complete? . . 4 What is an obvious sufficient statistic from an exponential family? . . 3 What is a curved exponential family? . . . . Exponential Family . . . . . . . . . . . . . . . . Review . . Summary Last Lecture . . 1 What is an exponential family distribution? 2 / 24 . . . . . . . . . .

. Summary February 5th, 2013 Biostatistics 602 - Lecture 08 Hyun Min Kang n n . k Theorem 6.2.25 3 / 24 . Review . . . . . . . . . . . . . . . . Exponential Family . . . . . . . . . . Suppose X 1 , · · · , X n is a random sample from pdf or pmf f X ( x | θ ) where   ∑ f X ( x | θ ) = h ( x ) c ( θ ) exp w j ( θ ) t j ( x )   j =1 is a member of an exponential family. Then the statistic T ( X )   ∑ ∑ T ( X ) = t 1 ( X j ) , · · · , t k ( X j )   j =1 j =1 is complete as long as the parameter space Θ contains an open set in R k

• Decompose f X x • Apply Theorem 6.2.10 to obtain a sufficient statistic and see if it is • Apply Theorem 6.2.25 to show that it is complete. • Apply Theorem 6.2.28 to show that it is minimal sufficient. . n . How to solve it . . . . . . . . i in the form of an an exponential family. equivalent to or related to T X and T X . Hyun Min Kang Biostatistics 602 - Lecture 08 February 5th, 2013 . 4 / 24 n . . . Problem . Exponential Family Example Summary Review i.i.d. Exponential Family . . . . . . . . . . . . . . . . whether (1) sufficient (2) complete, and (3) minimal sufficient. . . . . . . . . . . ∼ N ( µ, σ 2 ) . Determine whether the following statistics are X 1 , · · · , X n ( n ) ( ) ∑ ∑ ∑ X 2 X , s 2 ( X i − X ) 2 /( n − 1) T 1 ( X ) = , T 2 ( X ) = X = X i , i =1 i =1 i =1

• Apply Theorem 6.2.10 to obtain a sufficient statistic and see if it is • Apply Theorem 6.2.25 to show that it is complete. • Apply Theorem 6.2.28 to show that it is minimal sufficient. . . i n . How to solve it . equivalent to or related to T . whether (1) sufficient (2) complete, and (3) minimal sufficient. X and T X . Hyun Min Kang Biostatistics 602 - Lecture 08 February 5th, 2013 n 4 / 24 . . . . . . . . . . . . . . . Exponential Family Review . . Summary Exponential Family Example . Problem . . i.i.d. . . . . . . . . . . ∼ N ( µ, σ 2 ) . Determine whether the following statistics are X 1 , · · · , X n ( n ) ( ) ∑ ∑ ∑ X 2 X , s 2 ( X i − X ) 2 /( n − 1) T 1 ( X ) = , T 2 ( X ) = X = X i , i =1 i =1 i =1 • Decompose f X ( x | µ, σ ) in the form of an an exponential family.

• Apply Theorem 6.2.25 to show that it is complete. • Apply Theorem 6.2.28 to show that it is minimal sufficient. . i.i.d. February 5th, 2013 Biostatistics 602 - Lecture 08 Hyun Min Kang . . How to solve it . n i . n whether (1) sufficient (2) complete, and (3) minimal sufficient. 4 / 24 . . Review Summary . Exponential Family Exponential Family Example . . . . . . . . . . . . . Problem . . . . . . . . . . . . . . ∼ N ( µ, σ 2 ) . Determine whether the following statistics are X 1 , · · · , X n ( n ) ( ) ∑ ∑ ∑ X 2 X , s 2 ( X i − X ) 2 /( n − 1) T 1 ( X ) = , T 2 ( X ) = X = X i , i =1 i =1 i =1 • Decompose f X ( x | µ, σ ) in the form of an an exponential family. • Apply Theorem 6.2.10 to obtain a sufficient statistic and see if it is equivalent to or related to T 1 ( X ) and T 2 ( X ) .

• Apply Theorem 6.2.28 to show that it is minimal sufficient. . i.i.d. February 5th, 2013 Biostatistics 602 - Lecture 08 Hyun Min Kang . . How to solve it . n i . n whether (1) sufficient (2) complete, and (3) minimal sufficient. 4 / 24 . . . Exponential Family Summary . . . . . . . . . . . . . . Exponential Family Example . . Problem . Review . . . . . . . . . . ∼ N ( µ, σ 2 ) . Determine whether the following statistics are X 1 , · · · , X n ( n ) ( ) ∑ ∑ ∑ X 2 X , s 2 ( X i − X ) 2 /( n − 1) T 1 ( X ) = , T 2 ( X ) = X = X i , i =1 i =1 i =1 • Decompose f X ( x | µ, σ ) in the form of an an exponential family. • Apply Theorem 6.2.10 to obtain a sufficient statistic and see if it is equivalent to or related to T 1 ( X ) and T 2 ( X ) . • Apply Theorem 6.2.25 to show that it is complete.

. . February 5th, 2013 Biostatistics 602 - Lecture 08 Hyun Min Kang . . How to solve it . n i . n whether (1) sufficient (2) complete, and (3) minimal sufficient. i.i.d. 4 / 24 . . Exponential Family . . . . . . . . . . . . . . Summary Exponential Family Example . . . Problem Review . . . . . . . . . . ∼ N ( µ, σ 2 ) . Determine whether the following statistics are X 1 , · · · , X n ( n ) ( ) ∑ ∑ ∑ X 2 X , s 2 ( X i − X ) 2 /( n − 1) T 1 ( X ) = , T 2 ( X ) = X = X i , i =1 i =1 i =1 • Decompose f X ( x | µ, σ ) in the form of an an exponential family. • Apply Theorem 6.2.10 to obtain a sufficient statistic and see if it is equivalent to or related to T 1 ( X ) and T 2 ( X ) . • Apply Theorem 6.2.25 to show that it is complete. • Apply Theorem 6.2.28 to show that it is minimal sufficient.

. t x x t x x By Theorem 6.2.10, n i t X i n i X i w n i X n i X i T X is a sufficient statistic Hyun Min Kang Biostatistics 602 - Lecture 08 February 5th, 2013 t w . exp . . . . . . . . . . . . . . . . Exponential Family Review . Summary Applying Theorem 6.2.10 5 / 24 c h x where exp . . . . . . . . . . ( µ − µ 2 σ 2 x − x 2 1 ( ) ) f X ( x | µ, σ 2 ) = 2 πσ 2 exp 2 σ 2 2 σ 2

Data Reduction - Summary Lecture 08 Biostatistics 602 - Statistical - PowerPoint PPT Presentation

. . February 5th, 2013 Biostatistics 602 - Lecture 08 Hyun Min Kang February 5th, 2013 Hyun Min Kang Data Reduction - Summary Lecture 08 Biostatistics 602 - Statistical Inference . Summary . . Review Exponential Family . . . . . . .

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