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sta 2006 0708_chapter_11

sta 2006 0708_chapter_11 - STA 2006 Asymptotics of...

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STA 2006 Asymptotics of MLE (Section 6.14) 2007-2008 Term II 1 Maximum Likelihood Estimation Suppose the data X 1 , . . . , X n are randomly sampled from the p.d.f. or p.m.f. f ( x ; θ ) with the parameter θ being unknown but fixed. Define the likeli- hood function of the model as the joint p.d.f. or p.m.f. of the data and denote that by L ( θ ). That is, L ( θ ) = f ( x 1 , . . . , x n ; θ ). Let ˆ θ be the maximizer of L ( θ ). i.e., L ( ˆ θ ) = max θ Ω L ( θ ). The maximum likelihood estimate of θ is ˆ θ ( x 1 , . . . , x n ) (a real number) and the corresponding ˆ θ ( X 1 , . . . , X n ) (a random variable) is called the maximum likelihood estima- tor. Both of them are abbreviated as the MLE of θ . 2 Inference of MLE Question: How to test H 0 : θ = θ 0 vs. H 1 : θ > θ 0 ? For small n , the distribution of ˆ θ could be non-explicit. For large n , we could use CLT. Consider the likelihood function for i.i.d. data: L ( θ ) = f ( X 1 , . . . , X n ; θ ) = n Y i =1 f ( X i ; θ ) . Then, the corresponding log-likelihood function is given by: l ( θ ) = n X i =1 log( f ( X i ; θ )) . (1) Expand l 0 ( θ ) (the first derivative of l w.r.t. θ
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