MLE_Lecture 2 - Lecture 2 Examining the efficiency of estimators vis-vis the Cramer-Rao Lower Bound Hessian for N 2 model N 2 H= N xi i =1 4 H11 = i =1

# MLE_Lecture 2 - Lecture 2 Examining the efficiency of...

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Lecture 2 Examining the efficiency of estimators vis-à-vis the Cramer-Rao Lower Bound. Hessian for ) , ( 2 σ µ N model: = 2 2 2 2 2 2 2 2 2 2 2 2 2 6 2 1 = 4 4 1 = 4 1 = 2 ) ( ) , ; ( ) , ; ( ) , ; ( ) , ; ( ) ( 2 ) ( ) ( = σ σ µ µ σ σ µ σ µ σ µ µ σ µ σ µ σ σ µ σ µ σ x l x l x l x l x N x x N H i N i i N i i N i 0 < ˆ = 2 11 σ N H 4 2 6 2 1 = 4 4 1 = 4 1 = 2 2 ˆ 2 0 0 ˆ = ˆ ) ( ˆ 2 ˆ ) ( ˆ ) ( ˆ = ) ˆ , ˆ ( σ σ σ σ σ σ σ σ µ N N x x N x x x x N H i N i i N i i N i Then 0 > ˆ 2 = )] ˆ , ˆ ( [ 2 2 2 σ σ µ N H det , that is, the leading principal minors are changing signs. So H is negative-definite. Cramer-Rao Lower Bound (CRLB) { } 1 2 1 ) , ( = ] [ = σ µ I H E CRLB where ) , ( 2 σ µ I is the information matrix ) , ( = 2 0 0 = ) ( 2 ) ( ) ( = ] [ 2 4 2 6 2 1 = 4 4 1 = 4 1 = 2 σ µ σ σ σ µ σ σ µ σ µ σ I N N x N x x N E H E i N i i N i i N i

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• Spring '16
• Econometrics, Estimation theory, unbiased estimator, Minimum-variance unbiased estimator, Rao–Blackwell theorem, CRLB

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