Lecture9_412

# First the population cdf is now plugging in the

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Unformatted text preview: ulation pdf and cdf we have: when . Thus we know that , and its mean should be Therefore we have: That is, is an unbiased estimator of θ. i .i .d . Example 4. Y1 , , Yn ~ U [0, ] (a) Find the MOME for (b) Find the MLE for (c) Are the MOME and MLE unbiased estimators of θ? Solution: (a) f ( y ) 1 , 0 y 4 1 y2 12 E (Y ) y dy [ ] [ 0] 0 0 2 2 2 E (Y ) Y 2 ˆ Y 1 2Y n 1 (b) L f ( yi ) ( ) n , 0 y1 , , yn i 1 l nL n l o g 0 y1 , , yn , d ln L n ˆ 0 ? This is not good. d So, 0 y1 , , yn = > 0 y(1) , , y( n ) Y( n ) , L is maximized when Y( n ) ˆ The MLE for is Y 2 (n) Note: Example 4 is different from example 2 in that the likelihood L als o depends on the value of θ. The smaller the value of θ is, the larger the likelihood will be. Therefore, the MLE of θ is the smallest θ ˆ satisfying the inequality: . That value is . Thus the MLE is: 2 Y( n ) . (c) It is straight-forward to show that Thus the MOME is a...
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