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# ch07 - CHAPTER 7 Section 7-2 2n Xi E X 1 = E i =1 2n 2n = 1...

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CHAPTER 7 Section 7-2 7-1. E () µ = = = = = = n n X E n n X E X n i i n i i 2 2 1 2 1 2 2 1 2 1 1 E = = = = = = n n X E n n X E X n i i n i i 1 1 1 1 2 , 1 X and 2 X are unbiased estimators of µ . The variances are V 2n 2 1 σ = X and V n 2 2 = X ; compare the MSE (variance in this case), 2 1 2 / 2 / ) ˆ ( ) ˆ ( 2 2 2 1 = = = n n n n MSE MSE Θ Since both estimators are unbiased, examination of the variances would conclude that X 1 is the “better” estimator with the smaller variance. 7-2. E [] = = = + + + = ) 7 ( 7 1 )) ( 7 ( 7 1 ) ( ) ( ) ( 7 1 ˆ 7 2 1 1 X E X E X E X E " E = + = + + = ] 2 [ 2 1 ) ( ) ( ) 2 ( 2 1 ˆ 7 6 1 2 X E X E X E a) Both and are unbiased estimates of µ since the expected values of these statistics are 1 ˆ 2 ˆ equivalent to the true mean, µ . b) V 2 2 7 2 1 2 7 2 1 1 7 1 ) 7 ( 49 1 ) ( ) ( ) ( 7 1 7 ... ˆ = = + + + = + + + = X V X V X V X X X V " 7 ) ˆ ( 2 1 = V V )) ( ) ( ) ( 4 ( 4 1 ) ( ) ( ) 2 ( 2 1 2 2 ˆ 4 6 1 4 6 1 2 4 6 1 2 X V X V X V X V X V X V X X X V + + = + + = + = = ( ) 1 4 4 222 σσσ ++ = 1 4 6 2 σ 2 3 ) ˆ ( 2 2 = V Since both estimators are unbiased, the variances can be compared to decide to select the better estimator. The variance of is smaller than that of , is the better estimator. 1 ˆ 2 ˆ 1 ˆ 7-3. Since both and are unbiased, the variances of the estimators can be examined to determine the “better” estimator. The variance of is smaller than that of thus may be the better estimator. 1 ˆ 2 ˆ ± θ 2 1 ˆ θ ± θ 2 Relative Efficiency = 5 . 2 4 10 ) ˆ ( ) ( ) ˆ ( ) ˆ ( 2 1 2 1 = = = V V MSE MSE L 7-1

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7-4. Since both estimators are unbiased: Relative Efficiency = 21 2 2 / 3 7 / ) ˆ ( ) ˆ ( ) ˆ ( ) ˆ ( 2 2 2 1 2 1 = = Θ Θ = Θ Θ σ V V MSE MSE 7-5. 5 . 2 4 10 ) ˆ ( ) ˆ ( ) ˆ ( ) ˆ ( 2 1 2 1 = = = Θ V V MSE MSE 7-6. θ = ) ˆ ( 1 E 2 / ) ˆ ( 2 = E = ) ˆ ( 2 E Bias = θ θ 2 = θ 2 V = 10 V = 4 ) ˆ ( 1 ) ˆ ( 2 For unbiasedness, use since it is the only unbiased estimator. 1 ˆ As for minimum variance and efficiency we have: Relative Efficiency = 2 2 2 1 2 1 ) ) ˆ ( ( ) ) ˆ ( ( Bias V Bias V + + where bias for θ 1 is 0. Thus, Relative Efficiency = ( ) () 10 0 4 2 40 16 2 2 + + = + θ θ If the relative efficiency is less than or equal to 1, is the better estimator. 1 ˆ Use , when 1 ˆ 40 16 1 2 + θ 40 16 2 ≤+ θ 24 2 ≤θ θ ≤ − 4 899 . or θ ≥ 4899 . If −< then use . < . θ . 2 ˆ For unbiasedness, use . For efficiency, use when 1 ˆ 1 ˆ θ ≤ − 4 899 . or θ ≥ . and use when . 2 ˆ < .. θ 7-7. No bias = ) ˆ ( 1 E ) ˆ ( 12 ) ˆ ( 1 1 MSE V = = No bias = ) ˆ ( 2 E ) ˆ ( 10 ) ˆ ( 2 2 MSE V = = Bias [note that this includes (bias ) ˆ ( 3 E 6 ) ˆ ( 3 = MSE 2 )] To compare the three estimators, calculate the relative efficiencies: 2 . 1 10 12 ) ˆ ( ) ˆ ( 2 1 = = MSE MSE , since rel. eff. > 1 use as the estimator for θ 2 ˆ 2 6 12 ) ˆ ( ) ˆ ( 3 1 = = MSE MSE , since rel. eff. > 1 use as the estimator for θ 3 ˆ 8 . 1 6 10 ) ˆ ( ) ˆ ( 3 2 = = MSE MSE , since rel. eff. > 1 use as the estimator for θ 3 ˆ Conclusion: is the most efficient estimator with bias, but it is biased. is the best “unbiased” estimator.
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ch07 - CHAPTER 7 Section 7-2 2n Xi E X 1 = E i =1 2n 2n = 1...

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