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

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CHAPTER 4 Section 4-2 4-1. E ( ) ( ) XE X nn EX n n i i n i i n 1 = = == = = 1 2 1 2 2 1 2 1 2 2 µµ E ( ) ( ) X n n i i n i i n 2 = = = = 1 1 11 , X 1 and X 2 are unbiased estimators of µ . The variances are V ( ) X 1 2 2n = σ and V ( ) 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 () [] 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 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 and θ are unbiased estimates of µ since the expected values of these statistics are equivalent to ± 1 ± 2 the true mean, µ . b) V 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 )) 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 = 2 2 2 4 9 4 1 σ + σ + σ = ) 14 ( 4 1 2 σ 2 7 ) ˆ ( 2 2 σ = θ V Since both estimators are unbiased, the variances can be compared to decide which is the better estimator. The variance of is smaller than that of θ , is the ± θ 1 ± 2 ± θ 1 better estimator. 4-3. Since both θ and are unbiased, the variances of the estimators can be examined to determine which is the “better” estimator. The variance of is smaller than that of θ thus may be the better estimator. ± 1 ± θ 2 1 ˆ θ ± 2 ± θ 1 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( ± ) θθ 1 = ± )/ 2 2 = Bias E =− ( ± ) 2 = θ θ 2 = θ 2 V = 10 V = 4 ( ± ) θ 1 ( ± ) θ 2 For unbiasedness, use θ since it is the only unbiased estimator. As for minimum variance and efficiency we ± 1 have: Relative Efficiency = (( ± )) ± VB i a s i a s θ θ 1 2 1 2 2 2 + + 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. ± θ 1 Use θ , when ± 1 40 16 1 2 + θ 40 16 2 ≤+ θ 24 2 ≤θ θ ≤ − 4 899 .o r θ ≥ 4899 . If −< then use . < . θ . ± θ 2 For unbiasedness, use θ . For efficiency, use when ± 1 ± θ 1 θ ≤ − 4 899 r θ ≥ . and use when ± θ 2 –4.899 < θ < 4.899. 4-7. No bias ± ) θ 1 = θ θ θ VM S E ( ± )( ± ) 11 12 == No bias ± ) θ 2 = S E ( ± ± ) 22 10 Bias includes (bias ± ) θ 3 V( ± ) θ 3 6 = 2 ) To compare the three estimators, calculate the relative efficiencies: MSE MSE ( ± ) ( ± ) .
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This homework help was uploaded on 04/09/2008 for the course ENGR, STAT 320, 262, taught by Professor Harris during the Spring '08 term at Purdue.

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

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