But it is not a contradiction to the theorem because

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Unformatted text preview: e theorem. Because the domain, depends on . Thus the C-R Theorem doesn’t hold for this problem. i .i .d . Example 3. Let X1 , , X n ~ N ( , 2 ) , be a random sample from the normal population where both and 2 are unknown. Please derive (1) The maximum likelihood estimators for and 2 . (2) The best estimator for assuming that 2 is known. Solution: (1) MLEs for and 2 . The Likelihood function is: n n n i 1 1 L f ( xi ; , 2 ) 2 2 i 1 e ( xi ) 2 2 n 2 ( 1 2 2 ( xi )2 i 1 )e 2 2 n ln L (n)ln( 2 2 ) (x ) i 1 2 i 2 2 n X ln L ˆ 2 ( xi ) 0 i n i 1 n ln L 1 ( n ) 2 2 2 ( xi )2 i 1 4 2 n ˆ 0 2 ˆ ( xi )2 i 1 n n (x x ) i 1 2 i n (2) Use the Cramer-Rao lower bound: 4 1 f X ( , ) 2 2 2 e ( x )2 2 2 ( x )2 ln f ln( ) 2 2 2 2 d ln f ( x ) d 2 1 d 2 ln f 1 2 2 d Hence, the C-R lower bound of variance is 2 . n Since the normal pdf satisfies all the regularity conditions for the C-R lower bound theorem to hold, and since The variance of equals to th...
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This document was uploaded on 03/15/2014 for the course AMS 412 at SUNY Stony Brook.

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