ISM_chapter9 - Chapter 9: Properties of Point Estimators...

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181 Chapter 9: Properties of Point Estimators and Methods of Estimation 9.1 Refer to Ex. 8.8 where the variances of the four estimators were calculated. Thus, eff( 1 ˆ θ , 5 ˆ θ ) = 1/3 eff( 2 ˆ θ , 5 ˆ θ ) = 2/3 eff( 3 ˆ θ , 5 ˆ θ ) = 3/5. 9.2 a. The three estimators a unbiased since: E ( 1 ˆ μ ) = () μ = μ + μ = + ) ( ) ( ) ( 2 1 2 1 2 1 Y E Y E μ = μ + μ + μ = μ 4 / ) 2 ( 2 ) 2 ( 4 / ) ˆ ( 2 n n E μ = = μ ) ( ) ˆ ( 3 Y E E . b. The variances of the three estimators are 2 2 1 2 2 4 1 1 ) ( ) ˆ ( σ = σ + σ = μ V ) 2 ( 4 8 / 16 / ) 2 ( 4 ) 2 ( 16 / ) ˆ ( 2 2 2 2 2 2 2 σ + σ = σ + σ + σ = μ n n n V n V / ) ˆ ( 2 3 σ = μ . Thus, eff( 3 ˆ μ , 2 ˆ μ ) = ) 2 ( 8 2 n n , eff( 3 ˆ μ 1 ˆ μ ) = n /2. 9.3 a. E ( 1 ˆ θ ) = E ( Y ) – 1/2 = θ + 1/2 – 1/2 = θ . From Section 6.7, we can find the density function of 2 ˆ θ = Y ( n ) : 1 ) ( ) ( θ = n n y n y g , θ y θ + 1. From this, it is easily shown that E ( 2 ˆ θ ) = E ( Y ( n ) ) – n /( n + 1) = θ . b. V ( 1 ˆ θ ) = V ( Y ) = σ 2 / n = 1/(12 n ). With the density in part a , V ( 2 ˆ θ ) = V ( Y ( n ) ) = 2 ) 1 )( 2 ( + + n n n . Thus, eff( 1 ˆ θ , 2 ˆ θ ) = 2 2 ) 1 )( 2 ( 12 + + n n n . 9.4 See Exercises 8.18 and 6.74. Following those, we have that V ( 1 ˆ θ ) = ( n + 1) 2 V ( Y ( n ) ) = 2 2 θ + n n . Similarly, V ( 2 ˆ θ ) = 2 1 n n + V ( Y ( n ) ) = 2 ) 2 ( 1 θ + n n . Thus, the ratio of these variances is as given. 9.5 From Ex. 7.20, we know S 2 is unbiased and V ( S 2 ) = V ( 2 1 ˆ σ ) = 1 2 4 σ n . For 2 2 ˆ σ , note that Y 1 Y 2 is normal with mean 0 and variance σ 2 . So, 2 2 2 1 2 ) ( σ Y Y is chi–square with one degree of freedom and E ( 2 2 ˆ σ ) = σ 2 , V ( 2 2 ˆ σ ) = 2 σ 4 . Thus, we have that eff( 2 1 ˆ σ , 2 2 ˆ σ ) = n – 1. 9.6 Both estimators are unbiased and V ( 1 ˆ λ ) = λ /2 and V ( 2 ˆ λ ) = λ / n . The efficiency is 2/ n . 9.7 The estimator 1 ˆ θ is unbiased so MSE( 1 ˆ θ ) = V ( 1 ˆ θ ) = θ 2 . Also, 2 ˆ θ = Y is unbiased for θ ( θ is the mean) and V ( 2 ˆ θ ) = σ 2 / n = θ 2 / n . Thus, we have that eff( 1 ˆ θ , 2 ˆ θ ) = 1/ n .
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182 Chapter 9: Properties of Point Estimators and Methods of Estimation Instructor’s Solutions Manual 9.8 a. It is not difficult to show that 2 2 2 1 ) ( ln σ μ = y f , so I ( μ ) = σ 2 / n , Since V ( Y ) = σ 2 / n , Y is an efficient estimator of μ . b. Similarly, 2 2 2 ) ( ln λ λ = y y p and E (– Y / λ 2 ) = 1/ λ . Thus, I ( λ ) = λ / n . By Ex. 9.6, Y is an efficient estimator of λ . 9.9 a. X 6 = 1. b. - e. Answers vary. 9.10 a. - b. Answers vary. 9.11 a. - b. Answers vary. c. The simulations are different but get close at n = 50. 9.12 a. - b. Answers vary. 9.13 a. Sequences are different but settle down at large n . b. Sequences are different but settle down at large n . 9.14 a. the mean, 0. b. - c. the variability of the estimator decreases with n . 9.15 Referring to Ex. 9.3, since both estimators are unbiased and the variances go to 0 with as n goes to infinity the estimators are consistent. 9.16 From Ex. 9.5, V ( 2 2 ˆ σ ) = 2 σ 4 which is constant for all n . Thus, 2 2 ˆ σ is not a consistent estimator. 9.17 In Example 9.2, it was shown that both X and Y are consistent estimators of μ 1 and μ 2 , respectively. Using Theorem 9.2, X Y is a consistent estimator of μ 1 μ 2 .
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This note was uploaded on 03/18/2009 for the course STA 4321 taught by Professor Staff during the Spring '08 term at University of Florida.

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ISM_chapter9 - Chapter 9: Properties of Point Estimators...

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