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stat400lec25 - Fall 2005 1-Example X1 X2,… Xn is a random...

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Statistics 400 Section 7.1 Method of moments Review : We consider random variables for which the function form of pdf is known, but of the parameter of the pdf, say θ , is unknown. Parameter space : all possible values of θ . Estimator: The function of X1, X2,…, Xn used to estimate θ , say the statistics u( X1, X2,…, Xn), is called an estimator of θ . There are two general methods (1) Maximum likelihood (2) Method of moments Likelihood function : multiplication of the pmf or pdf. L ( θ )=f( x 1 ; θ )f( x 2 ; θ )...f( x n ; θ ) Maximize likelihood: Maximize L ( θ ) with respect to θ 0 ) ( = θ θ d dL Unbiased estimator: If E[u(X1, X2,…, Xn)]= θ , the statistics u(X1, X2,…, Xn) is called an unbiased estimator of θ . Ping Ma
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Unformatted text preview: Fall 2005- 1 -Example : X1, X2,…, Xn is a random sample of size n from the normal distribution N( μ , σ 2 ). Are the maximum likelihood estimators of μ and σ 2 unbiased estimators? Ping Ma Lecture 25 Fall 2005- 2 -Method of moment: 1 st sample moment = 1 st moment 2 nd sample moment=2 nd moment (If there are two parameters) and 3 rd sample moment=3 rd moment so on ( If there are more than two parameters) Example : X1, X2,…, Xn is a random sample of size n from the normal distribution N( μ , σ 2 ). Find method of moment estimators of μ and σ 2 . Ping Ma Lecture 25 Fall 2005- 3 -Ping Ma Lecture 25 Fall 2005- 4 -...
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