20props - EECS 501 ESTIMATOR PROPERTIES Fall 2001 Problem:...

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EECS 501 ESTIMATOR PROPERTIES Fall 2001 Problem: Let { x 1 ...x N } be iidrv with x i N ( m,σ 2 ) and m, σ 2 unknown . Want: To compute ˆ m MLE and ˆ σ 2 MLE based on observations { X 1 ...X N } . Solution: f x 1 ...x N ( X 1 ...X N ) = Q N i =1 1 2 πσ 2 e - 1 2 ( X i - m ) 2 2 since x i indpt rvs. Set: 0 = ∂m log f x 1 ...x N = ∂m [ - N 2 log(2 π ) - N 2 log σ 2 - 1 2 N i =1 ( X i - m ) 2 2 ] = 1 σ 2 N i =1 ( X i - m ) = 0 ˆ m MLE = 1 N N i =1 X i = samplemean . Set: 0 = ∂σ 2 log f x 1 ...x N = ∂σ 2 [ - N 2 log(2 π ) - N 2 log σ 2 - 1 2 N i =1 ( X i - m ) 2 2 ] = - N 2 1 σ 2 + 1 2 N i =1 ( X i - m ) 2 / ( σ 2 ) 2 = 0 ˆ σ 2 MLE = 1 N N i =1 ( X i - m ) 2 . Replace m in ˆ σ 2 MLE with ˆ m MLE ˆ σ 2 MLE = samplevariance . Note: ˆ σ MLE = p ˆ σ 2 MLE : MLE commutes with nonlinear functions g ( a ). Why?
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