Lecture 18-2007

# Lecture 18-2007 - Mean Squared Error and Maximum Likelihood...

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Mean Squared Error and Maximum Likelihood Lecture XVIII

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Mean Squared Error As stated in our discussion on closeness, one potential measure for the goodness of an estimator is ( 29 - 2 ˆ θ θ E
In the preceding example, the mean square error of the estimate can be written as: where θ is the true parameter value between zero and one. ( 29 [ ] 2 θ - T E

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This expected value is conditioned on the probability of T at each level value of θ . [ ] ( 29 X X X P - - = 1 1 , θ θ θ [ ] ( 29 2 1 2 1 1 2 1 1 , , X X X X X X P - - + - = θ θ θ ( 29 [ ] ( 29 [ ] ( 29 [ ] ( 29 2 2 2 1 , 1 , 1 5 . , 1 , 0 2 0 , 0 , 0 θ θ θ θ θ θ θ - + - + - = P P P MSE
( 29 [ ] ( 29 [ ] ( 29 2 2 1 , 1 0 , 0 θ θ θ θ θ - + - = P P MSE ( 29 2 ) 5 (. θ θ - = MSE

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MSEs of Each Estimator 0.2 0.4 0.6 0.8 1 0.05 0.1 0.15 0.2 0.25
Definition 7.2.1. Let X and Y be two estimators of θ . We say that X is better (or more efficient) than Y if E ( X - θ ) 2 E ( Y - θ ) for all θ in Θ and strictly less than for at least one θ in Θ .

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When an estimator is dominated by another estimator, the dominated estimator is inadmissable. Definition 7.2.2. Let θ be an estimator of θ. We say that θ is inadmissible if there is
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