Problem3_28 - STA 6934 Problem 3.28 (a,b,c) Vladimir L....

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STA 6934 Problem 3.28 (a,b,c) Vladimir L. Boginski We need to evaluate () ( ) a P xa fx d x >= ò , Xf : . a) We have the following estimators: 1 1 1 , , n ii i IX a X fi i d n δ = => å : 3 1 1 n i i IX n µ = å , where PX > is known, and a > . The control variate estimator: 2 11 1 [ ( ) ] nn IX a n βµ == + > > åå . Since 2 21 3 1 3 var( ) var( ) var( ) 2 cov( , ) δδ β =+ + ,and 13 1 cov( , ) ( )[1 ( )] PX a n > , we can find 3 var( ) : 3 2 1 1 var( ) var( ( )) n i i n = å , (because X i are iid). 22 2 var ( ) [ ( )] [ ( ( ))] [( ) ] [ (( ) ) ] [ () ] [ ] [ 1 ] i EIX fxd x x µµ ∞∞ > > = > = = >=> −> òò (We used the fact that 2 [ ( )] [ ( )] EI X > .) So, 3 2 var( ) ( ( )] ( ( )] n P xP x P x = >−> = . b) 2 3 1 3 var( ) var( ) var( ) 2 cov( , ) + . If we want 2 to improve 1 (in the sense of reducing variance) , we need the following condition to be satisfied: 2 31 3 var( ) 2 cov( , ) 0 βδβ +< .
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Since 2 3 var( ) 0 βδ ,and 13 cov( , ) 0 δδ (see part a), we need 0 β < . Also from the inequality 2 31 3 var( ) 2 cov( , ) 0 βδβ δ +< we get 2 3 3 var( ) 2 cov( , ), cov( , ) 2 var( ) <− −< or 3 cov( ,
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Problem3_28 - STA 6934 Problem 3.28 (a,b,c) Vladimir L....

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