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hw3sol - 36-226 Summer 2010 Homework 3 Solutions 1 X1 Xn...

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36-226 Summer 2010 Homework 3 Solutions 1. X 1 , . . . , X n are iid ∼ U (0 , 2 θ ). Then f X ( x ) = 1 2 θ I [0 , 2 θ ] ( x ) L ( θ ) = 1 (2 θ ) n I [0 , 2 θ ] ( x 1 , . . . , x n ) Note that you can not find the mle just by direct calculation in this case (which you can usually realize if you actually try solving it that way). So, you have to find the mle by inspection, like we did in class for the case of the U (0 , θ ) distribution. First, you note that in general L ( θ ) decreases as θ increases ( θ ↑ ⇒ (2 θ ) n 1 (2 θ ) n ). Also, we need to look at the values for which θ is non-zero. Since L ( θ ) = 1 (2 θ ) n I [0 , 2 θ ] ( x 1 , . . . , x n ) = 1 (2 θ ) n I [0 , 2 θ ] ( x ( n ) ) L ( θ ) is non-zero if x i 2 θ, i = 1 , . . . , n ⇐⇒ 2 θ x ( n ) (the maximum) ⇐⇒ θ x ( n ) 2 . So, the likelihood L ( θ ) looks like this: X ( n ) 2 is the MLE. Now, we also need to find the expectation of the mle. 1
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E [ b θ ] = E X ( n ) 2 = 1 2 E [ X ( n ) ] = To find the expectation of the maximum, we first find its pdf using the result about the distribution of order statistics.
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