First the population cdf is now plugging in the

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online