Dr. Hackney STA Solutions pg 93

Dr. Hackney STA Solutions pg 93 - Second Edition 6-7 6.29...

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Second Edition 6-7 6.29 Let f j = logistic( α j j ), j = 0 , 1 ,...,k . From Theorem 6 . 6 . 5, the statistic T ( x ) = ±Q n i =1 f 1 ( x i ) Q n i =1 f 0 ( x i ) ,..., Q n i =1 f k ( x i ) Q n i =1 f 0 ( x i ) ² = ±Q n i =1 f 1 ( x ( i ) ) Q n i =1 f 0 ( x ( i ) ) ,..., Q n i =1 f k ( x ( i ) ) Q n i =1 f 0 ( x ( i ) ) ² is minimal sufficient for the family { f 0 ,f 1 ,...,f k } . As T is a 1 - 1 function of the order statistics, the order statistics are also minimal sufficient for the family { f 0 ,f 1 ,...,f k } . If F is a nonparametric family, f j ∈ F , so part ( b ) of Theorem 6 . 6 . 5 can now be directly applied to show that the order statistics are minimal sufficient for F . 6.30 a. From Exercise 6.9b, we have that X (1) is a minimal sufficient statistic. To check completeness compute f Y 1 ( y ), where Y 1 = X (1) . From Theorem 5.4.4 we have f Y 1 ( y ) = f X ( y ) (1 - F X ( y )) n - 1 n = e - ( y - μ ) h e - ( y - μ ) i n - 1 n = ne - n ( y - μ ) , y > μ. Now, write E
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.

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