lec12 - Lecture 12 Let us give several more examples of...

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Unformatted text preview: Lecture 12 Let us give several more examples of finding sufficient statistics. Example 1. Poisson Distribution ( ) has p.f. f ( x | ) = x x ! e- for x = 0 , 1 , 2 , . . . and the joint p.f. is f ( x 1 , , x n | ) = P x i Q n i =1 x i ! e- n = 1 Q n i =1 X i ! e- n P X i . Therefore we can take u ( x 1 , . . . , x n ) = 1 Q n i =1 X i ! , T ( x 1 , . . . , x n ) = n X i =1 x i and v ( T, ) = e- n T . Therefore, by Neyman-Fisher factorization criterion T = n i =1 X i is a sufficient statis- tics. Example 2. Consider a family of normal distributions N ( , 2 ) and assume that 2 is a given known parameter and is the only unknown parameter of the family. The p.d.f. is given by f ( x | ) = 1 2 e- ( x- ) 2 2 2 and the joint p.d.f. is f ( x 1 , . . . , x n | ) = 1 ( 2 ) n exp n- n X i =1 ( x i- ) 2 2 2 o = 1 ( 2 ) n exp n- x 2 i 2 2 + x i 2- n 2 2 2 o = 1 ( 2 ) n exp n- x 2 i 2 2 o exp n X x i 2- n 2 2 2 o . 45 LECTURE 12. 46 If we take T = n i =1 X i , u ( x 1 , . . . , x n ) = 1 ( 2 ) n exp n- x 2 i 2 2 o and v ( T, ) = exp n T 2- n 2 2 2 o , then Neyman-Fisher criterion proves that T is a sufficient statistics....
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lec12 - Lecture 12 Let us give several more examples of...

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