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

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Unformatted text preview: Mean Squared Error and Maximum Likelihood Lecture XVIII 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 • 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 , 2 , , θ θ θ θ θ θ θ- +- +- = P P P MSE ( 29 [ ] ( 29 [ ] ( 29 2 2 1 , 1 , θ θ θ θ θ- +- = P P MSE ( 29 2 ) 5 (. θ θ- = MSE 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 . • When an estimator is dominated by another estimator, the dominated estimator is inadmissable.inadmissable....
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This note was uploaded on 07/18/2011 for the course AEB 6933 taught by Professor Carriker during the Fall '09 term at University of Florida.

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

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