sta 2006 1011_chap10

sta 2006 1011_chap10 - Theory of Statistical Inference Dr....

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Unformatted text preview: Theory of Statistical Inference Dr. Phillip YAM 2010/2011 Spring Semester Reference: Chapter 10: Section 7 of “Some Theory” by Hogg and Tanis. Section 10.7 ASYMPTOTIC DISTRIBUTIONS OF MAXIMUM LIKELIHOOD ESTIMATORS I Recall that the maximum likelihood estimator b θ is obtained by solving ∂ [ln L ( b θ )] ∂θ = 0 , I By applying Taylor series expansion up to first order: ∂ [ln L ( θ )] ∂θ + ( b θ- θ ) ∂ 2 [ln L ( θ )] ∂θ 2 ≈ , I This approximation is good enough only if b θ is close to θ , an adequate mathematical proof involves those conditions, which we have not given here. (See Hogg, McKean, and Craig, 2005.) b θ- θ = ∂ [ln L ( θ )] ∂θ- ∂ 2 [ln L ( θ )] ∂θ 2 . Section 10.7 ASYMPTOTIC DISTRIBUTIONS OF MAXIMUM LIKELIHOOD ESTIMATORS I ∂ ln L ( θ ) ∂θ = n X i =1 ∂ [ln f ( X i ; θ )] ∂θ , I Y i = ∂ [ln f ( X i ; θ )] ∂θ , I Z ∞-∞ ∂ [ln f ( x ; θ )] ∂θ f ( x ; θ ) dx = Z ∞-∞ ∂ [ f ( x ; θ )] ∂θ f ( x ; θ ) f (...
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sta 2006 1011_chap10 - Theory of Statistical Inference Dr....

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