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st422%20hw05%20solution

st422%20hw05%20solution - Solutions Homework 5 Q 9.2 a ^...

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Solutions: Homework 5 1 1 2 2 3 1 2 1 3 2 2 2 1 2 2 2 2 2 2 2 9.2: a) ˆ E( ) [( )/ 2] 2 / ... ( 2) ˆ E( ) [(1/ 4) (1/ 4) ] (1/ 4) (1/ 4) 2( 2) 2( 2) ˆ E( ) ( ) ( / ) ) ˆ ( ) ( )/ 4 / 2 ( 2) ˆ ( ) ( /16) ( /16) ( /8) 4( 2) 4 n n Q E Y Y Y Y Y n E Y Y n n E Y E n n b V n V n μ μ μ μ μ μ μ μ μ μ μ μ μ σ σ σ σ σ μ σ σ σ - = + = = + + - = + + = + + = - - = = = = + = - = + + = + - 2 3 1 1 3 2 2 2 2 3 1 1 3 2 ( 2) ( / 2) ˆ ˆ ˆ ˆ Efficiency of relative to is V( )/ ( ) / 2 ( / ) ( /8) 4( 2) ˆ ˆ ˆ ˆ Efficiency of relative to is V( )/ ( ) ( /8) ( / ) 4( 2) 8( 2) n V n n n n n V n n n n σ μ μ μ μ σ σ σ μ μ μ μ σ - = = + - = = + = - -

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1 2 1 2 ( ) 1 2 1 1 9.3: Given Y , ,... form a random sample from from U( , 1) ˆ (1/ 2) ˆ ( /( 1)) a) ˆ ˆ To show that and are unbiased for ˆ ( ) ( 0.5) [ ( 1)]/ 2 (1/ 2) ˆ Thus, is an unbiased est n n Q Y Y Y Y n n E E Y θ θ θ θ θ θ θ θ θ θ θ θ + = - = - + = - = + + - = ( ) 1 n-1 imator for ( , 1) ( ) 1 y +1 =0 Otherwise ( ) ( ) 1 ( ) 0 y< =y- y +1 =1 y> +1 ( ) ( ) ( ) =n(y- ) y n i y n y Y s U f y F y P Y y du y F y g y nF y f y θ θ θ θ θ θ θ θ θ θ θ θ θ θ - + = = = = - = = ( ) 2 ( ) 2 1 2 ( ) ( ) +1 = 0 otherwise ( ) ( ,1) ˆ E( ) [( ) ] ( /( 1)) [ /( 1)] [ /( 1)] ˆ Thus is an unbiased estimator for ) ˆ ( ) ( ) (1/12 ) ˆ ( ) ( ) ( n n n n Y Beta n E Y n n n n n n b V V Y n V V Y V Y θ θ θ θ θ θ θ θ θ θ θ θ - = - + - - = + + - + = = = = = - + ( ) 2 2 1 2 2 1 2 ) ( ) ( 1) ( 2) 12 ˆ ˆ ˆ ˆ eff( , ) ( )/ ( ) ( 1) ( 2) n n V Y n n n V V n n θ θ θ θ θ θ = - = + + = = + +
(1) (1) (1) 1 2 ( ) 1 2 1 (1) 2 ( ) ( 1) ( 1) (1) (1) ( 1) (1) 9.4: min( , ... ) max( , ... ) (0, ) ˆ ˆ ( 1) and [( 1)/ ] ( ) ( )[1 ( )] ( / )[1 ( / )] 0 / ( ) (1 ) 0 U (1, ) n n n i n n n Y n U Q Y Y Y Y Y Y Y Y Y s U n Y n n Y f y nf y F y n y y U Y f u n u u Beta n θ θ θ θ θ

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