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Lecture18-2010 - Mean Squared Error and Maximum Likelihood...

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Mean Squared Error and Maximum Likelihood: Lecture XVIII Charles B. Moss October 12, 2010 I. Mean Squared Error A. As stated in our discussion on closeness, one partential measure for the goodness of an estimator is E ˆ θ θ 2 (1) B. In the preceding example, the mean square error of the estimate can be written as E ( T θ ) 2 (2) where θ is the true parameter value between zero and one. C. This expected value is conditioned on the probability of T at each level value of θ . For example, if θ = 0 then the probability of each X becomes P [ X, θ ] = θ X (1 θ ) 1 X (3) If the two events are independent P [ X 1 , X 2 , θ ] = θ X 1 + X 2 (1 θ ) 1 X 1 X 2 (4) The mean squared error at any theta can then be derived as MSE ( θ ) = P [0 , 0 , θ ] (0 θ ) 2 +2 P [0 , 1 , θ ] (0 . 5 θ ) 2 + P [1 , 1 , θ ] (1 θ ) 2 . (5) 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XVIII Fall 2010 Figure 1: Comparison of MSE for Various Estimators D. The mean squared error for S can similarly be computed as MSE ( θ ) = P [0 , θ ] (0 θ ) 2 + P [1 , θ ] (1 θ ) 2 (6) E. Finally, the mean square error of W can be written as MSE ( θ ) = (0 . 5 θ ) 2 (7) F. The mean squared error for each estimator is presented in Figure 1. G. 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 ) 2 for all θ Θ and strictly less than for at least one θ Θ.
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