# ch04 - CHAPTER 4 Section 4-2 2n Xi 1 2n 1 X1 = E i =1 = E...

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CHAPTER 4 Section 4-2 4-1. E ( 29 ( 29 X E X n n E X n n i i n i i n 1 = =  = = = = 1 2 1 2 2 1 2 1 2 2 μ μ E ( 29 ( 29 X E X n n E X n n i i n i i n 2 = =  = = = = 1 1 1 1 μ μ , X 1 and X 2 are unbiased estimators of μ . The variances are V ( 29 X 1 2 2n = σ and V ( 29 X 2 2 n = σ ; compare the MSE (variance in this case), MSE MSE n n n n ( ) ( ) / / θ θ σ σ 1 2 2 2 2 2 1 2 = = = Since both estimators are unbiased, examination of the variances would conclude that X 1 is the “better” estimator with the smaller variance. 4-2. E ( 29 [ ] μ μ θ = = = + + + = ) 9 ( 9 1 )) X ( E 9 ( 9 1 ) X ( E ) X ( E ) X ( E 9 1 ˆ 9 2 1 1 E ( 29 [ ] μ = μ + μ - μ = + - = θ ] 2 3 [ 2 1 ) X 2 ( E ) X ( E ) X 3 ( E 2 1 ˆ 4 6 1 2 a) Both θ 1 and θ 2 are unbiased estimates of μ since the expected values of these statistics are equivalent to the true mean, μ . b) V ( 29 ( 29 2 2 9 2 1 2 9 2 1 1 9 1 ) 9 ( 81 1 ) X ( V ) X ( V ) X ( V 9 1 9 X ... X X V ˆ σ = σ = + + + = + + + = θ 9 ) ˆ ( V 2 1 σ = θ V ( 29 ( 29 )) X ( V 4 ) X ( V ) X ( V 9 ( 4 1 ) X 2 ( V ) X ( V ) X 3 ( V 2 1 2 X 2 X X 3 V ˆ 4 6 1 4 6 1 2 4 6 1 2 + + = + + = + - = θ = ( 29 2 2 2 4 9 4 1 σ + σ + σ = ) 14 ( 4 1 2 σ 2 7 ) ˆ ( V 2 2 σ = θ Since both estimators are unbiased, the variances can be compared to decide which is the better estimator. The variance of θ 1 is smaller than that of θ 2 , θ 1 is the better estimator. 4-3. Since both θ 1 and θ 2 are unbiased, the variances of the estimators can be examined to determine which is the “better” estimator. The variance of 1 ˆ θ is smaller than that of θ 2 thus θ 1 may be the better estimator. Relative Efficiency = 5 . 0 4 2 ) ˆ ( V ) ˆ ( V ) ˆ ( MSE ) ˆ ( MSE 2 1 2 1 = = θ θ = θ θ 1

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4-4. Since both estimators are unbiased: Relative Efficiency = 63 2 2 / 7 9 / ) ˆ ( V ) ˆ ( V ) ˆ ( MSE ) ˆ ( MSE 2 2 2 1 2 1 = = = σ σ θ θ θ θ 4-5. 5 . 0 4 2 ) ˆ ( V ) ˆ ( V ) ˆ ( MSE ) ˆ ( MSE 2 1 2 1 = = θ θ = θ θ 4-6. E( ) θ θ 1 = E( ) / θ θ 2 2 = Bias E = - ( ) θ θ 2 = θ θ 2 - = - θ 2 V ( ) θ 1 = 10 V ( ) θ 2 = 4 For unbiasedness, use θ 1 since it is the only unbiased estimator. As for minimum variance and efficiency we have: Relative Efficiency = ( ( ) ) ( ( ) ) V Bias V Bias θ θ 1 2 1 2 2 2 + + where, Bias for θ 1 is 0. Thus, Relative Efficiency = ( 29 ( ) 10 0 4 2 40 16 2 2 + + - = + θ θ If the relative efficiency is less than or equal to 1, θ 1 is the better estimator.
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ch04 - CHAPTER 4 Section 4-2 2n Xi 1 2n 1 X1 = E i =1 = E...

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