HW 4 Solutions

HW 4 Solutions - HW 4 Solutions 8.8 ^ ^ ^ ^ a. Note that 1...

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HW 4 Solutions 8.8 a. Note that 1 ˆ θ , 2 ˆ θ , 3 ˆ θ and 5 ˆ θ are simple linear combinations of Y 1 , Y 2 , and Y 3 . So, it is easily shown that all four of these estimators are unbiased . More explicitly, consider . Then From Ex. 6.81 it was shown that 4 ˆ θ has an exponential distribution with mean θ/3, so this estimator is biased. b. It is easily shown that V ( 1 ˆ θ ) = θ 2 , V ( 2 ˆ θ ) = θ 2 /2, V ( 3 ˆ θ ) = 5θ 2 /9, and V ( 5 ˆ θ ) = θ 2 /9, so the estimator 5 ˆ θ is unbiased and has the smallest variance. 8.15 Using standard techniques, it can be shown that E ( Y ) =(3/2)β, E ( Y 2 ) = 3β 2 . Also it is easiliy shown that Y (1) follows the Pareto family with density function ) 1 3 ( 3 ) 1 ( 3 ) ( + - β = n n y n y g , y ≥ β. Thus, E ( Y (1) ) = ( 29 β - 1 3 3 n n and 2 2 3 3 2 ) 1 ( ) ( β = - n n Y E . a. With ) 1 ( ˆ Y = β , ( 29 ( 29 β = β - β =
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This note was uploaded on 10/02/2009 for the course PSTAT 5E taught by Professor Eduardomontoya during the Spring '08 term at UCSB.

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