# Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat 312: Lecture 05Maximum Likelihood EstimationMoo K. Chung[email protected]September 14, 20041.(Invariance Principle) Ifˆθis the MLE’s ofparameterθthen the MLE ofh(θ)ish(ˆθ)forsome functionh.Proof(partial). Consider likelihood functionL(θ).ˆθsatisfiesdL(θ)= 0. Letφ=h(θ).Then the likelihood function forφ=h(θ)is given byL(h-1(φ)).Differentiating thelikelihood with respect toφ, we haveL(h-1(φ))=dL(θ)=dL(θ)1h0(θ)= 0.2.Loglikelihood.MaximizingL(θ)is equiva-lent to maximizinglnL(θ)sincelnis an in-creasing function.Example.This technique is best illus-trated by finding the MLE of parameters inN(μ, σ2).ˆμ=¯X,ˆσ2=1nnXi=1(Xi-¯X)2are the MLE ofμandσ2respectively. Notethatˆσ2is not un unbiased estimator ofσ2.3.Asymptotic unbiasness.When the samplesize is large, the maximum likelihood esti-mator ofθis approximately unbiased. TheMLE ofθ
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Term
Fall
Professor
Chung
Tags
Maximum likelihood, Likelihood function, maximum likelihood estimator