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Lecture 20-2007

# 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 , σ μ πσ σ μ ( 29 ( 29 ( 29 ∑ =--- = n i i X n L 1 2 2 2 2 1 ln 2 ln μ σ σ This expression can be solved for the optimal choice of σ 2 by differentiating with respect to σ 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 ˆ 2 1 2 ln μ σ μ σ μ σ σ σ 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- -- =--- -- = ∑ ∑ ∑ ∑ = = = = μ μ μ μ I ntuitively, the maximum likelihood estimate of μ 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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