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Unformatted text preview: 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 : θ = θ vs. H 1 : θ > θ ? 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 ) ....
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This note was uploaded on 06/05/2011 for the course STATISTICS 2006 taught by Professor Ho during the Spring '11 term at CUHK.
- Spring '11