ch6sol - Chapter 6 Principles of Data Reduction 6.1 By the...

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Unformatted text preview: Chapter 6 Principles of Data Reduction 6.1 By the Factorization Theorem, | X | is sufficient because the pdf of X is f ( x | 2 ) = 1 2 e- x 2 / 2 2 = 1 2 e-| x | 2 / 2 2 = g ( | x || 2 ) 1 | {z } h ( x ) . 6.2 By the Factorization Theorem, T ( X ) = min i ( X i /i ) is sufficient because the joint pdf is f ( x 1 ,...,x n | ) = n Y i =1 e i- x i I ( i, + ) ( x i ) = e in I ( , + ) ( T ( x )) | {z } g ( T ( x ) | ) e- i x i | {z } h ( x ) . Notice, we use the fact that i > 0, and the fact that all x i s > i if and only if min i ( x i /i ) > . 6.3 Let x (1) = min i x i . Then the joint pdf is f ( x 1 ,...,x n | , ) = n Y i =1 1 e- ( x i- ) / I ( , ) ( x i ) = e / n e- i x i / I ( , ) ( x (1) ) | {z } g ( x (1) , i x i | , ) 1 | {z } h ( x ) . Thus, by the Factorization Theorem, ( X (1) , i X i ) is a sufficient statistic for ( , ). 6.4 The joint pdf is n Y j =1 h ( x j ) c ( ) exp k X i =1 w i ( ) t i ( x j ) ! = c ( ) n exp k X i =1 w i ( ) n X j =1 t i ( x j ) | {z } g ( T ( x ) | ) n Y j =1 h ( x j ) | {z } h ( x ) . By the Factorization Theorem, n j =1 t 1 ( X j ) ,..., n j =1 t k ( X j ) is a sufficient statistic for . 6.5 The sample density is given by n Y i =1 f ( x i | ) = n Y i =1 1 2 i I (- i ( - 1) x i i ( + 1)) = 1 2 n n Y i =1 1 i ! I min x i i - ( - 1) I max x i i + 1 . Thus (min X i /i, max X i /i ) is sufficient for . 6-2 Solutions Manual for Statistical Inference 6.6 The joint pdf is given by f ( x 1 ,...,x n | , ) = n Y i =1 1 ( ) x i - 1 e- x i / = 1 ( ) n n Y i =1 x i ! - 1 e- i x i / . By the Factorization Theorem, ( Q n i =1 X i , n i =1 X i ) is sufficient for ( , ). 6.7 Let x (1) = min i { x 1 ,...,x n } , x ( n ) = max i { x 1 ,...,x n } , y (1) = min i { y 1 ,...,y n } and y ( n ) = max i { y 1 ,...,y n } . Then the joint pdf is f ( x , y | ) = n Y i =1 1 ( 3- 1 )( 4- 2 ) I ( 1 , 3 ) ( x i ) I ( 2 , 4 ) ( y i ) = 1 ( 3- 1 )( 4- 2 ) n I ( 1 , ) ( x (1) ) I (- , 3 ) ( x ( n ) ) I ( 2 , ) ( y (1) ) I (- , 4 ) ( y ( n ) ) | {z } g ( T ( x ) | ) 1 |{z} h ( x ) . By the Factorization Theorem, ( X (1) ,X ( n ) ,Y (1) ,Y ( n ) ) is sufficient for ( 1 , 2 , 3 , 4 ). 6.9 Use Theorem 6.2.13. a. f ( x | ) f ( y | ) = (2 )- n/ 2 e- i ( x i- ) 2 / 2 (2 )- n/ 2 e- i ( y i- ) 2 / 2 = exp- 1 2 " n X i =1 x 2 i- n X i =1 y 2 i ! +2 n ( y- x ) # . This is constant as a function of if and only if y = x ; therefore X is a minimal sufficient statistic for ....
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This note was uploaded on 04/18/2010 for the course STAT 622 taught by Professor Peruggia,m during the Spring '08 term at Ohio State.

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ch6sol - Chapter 6 Principles of Data Reduction 6.1 By the...

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