Lecture 20-2007 - Lectur e XX Concentrated Likelihood...

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Unformatted text preview: Lectur e XX Concentrated Likelihood Functions The more general form of the normal likelihood function can be written as: ( 29 ( 29 = -- = n i i X X L 1 2 2 2 2 2 exp 2 1 , ( 29 ( 29 ( 29 =--- = n i i X n L 1 2 2 2 2 1 ln 2 ln This expression can be solved for the optimal choice of 2 by differentiating with respect to 2 : ( 29 ( 29 ( 29 ( 29 ( 29 = = =- = =- +- =- +- = n i i MLE n i i n i i X n X n X n L 1 2 2 1 2 2 1 2 2 2 2 2 1 2 1 2 ln Substituting this result into the original logarithmic likelihood yields ( 29 ( 29 ( 29 ( 29 ( 29 2 1 ln 2 1 2 1 1 ln 2 ln 1 2 1 2 1 2 1 2 n X n n X X n X n n L n i i n i i n j j n i i- -- =--- -- = = = = = I ntuitively, the maximum likelihood estimate of is that value that minimizes the mean square error of the estimator. Thus, the least squares...
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Lecture 20-2007 - Lectur e XX Concentrated Likelihood...

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