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chap07PRN econ 325

A suggestion is to choose the estimator with minimum

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A suggestion is to choose the estimator with minimum variance. Let 1 ˆ θ and 2 ˆ θ be two unbiased estimators of the population parameter θ . That is, θ = θ ) ˆ ( E 1 and θ = θ ) ˆ ( E 2 1 ˆ θ is said to be more efficient than 2 ˆ θ if: ) ˆ ( Var ) ˆ ( Var 2 1 θ < θ The relative efficiency of one estimator with respect to another is the variance ratio: ) ˆ ( Var ) ˆ ( Var 1 2 θ θ h If θ ˆ is an unbiased estimator of θ , and no other unbiased estimator has smaller variance than θ ˆ , then θ ˆ is said to be the most efficient or minimum variance unbiased estimator of θ . Econ 325 – Chapter 7 6 Example Continued: For the random sample 1 X , 2 X , 3 X , introduced above, assume the population variance is 2 σ and also assume independence. That is, 2 3 2 1 ) X ( Var ) X ( Var ) X ( Var σ = = = and 0 ) X , X ( Cov ) X , X ( Cov ) X , X ( Cov 3 2 3 1 2 1 = = = Two unbiased estimators for the population mean were proposed as: ) X X X ( 3 1 X 3 2 1 + + = and ) X X 4 X ( 6 1 X 3 2 1 W + + = The variance of the first estimator is: 3 ) 3 ( 9 1 )] X ( Var ) X ( Var ) X ( Var [ 9 1 ) X ( Var 2 2 3 2 1 σ = σ = + + = The variance of the second estimator is: 2 ) 18 ( 36 1 )] X ( Var ) X ( Var 16 ) X ( Var [ 36 1 ) X ( Var 2 2 3 2 1 W σ = σ = + + =
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Econ 325 – Chapter 7 7 It can be seen that: ) X ( Var ) X ( Var W < Therefore, X is more efficient than W X . The relative efficiency is: 5 . 1 2 3 3 2 ) X ( Var ) X ( Var 2 2 W = = σ σ = Note: A reporting style for the measure of relative efficiency is to place the higher variance in the numerator. Econ 325 – Chapter 7 8 As a variation, suppose that the random sample 1 X , 2 X , 3 X have probability distributions with identical population mean μ but unequal population variances. Assume: 2 1 4 ) X ( Var σ = , 2 2 ) X ( Var σ = and 2 3 4 ) X ( Var σ = The revised variances for the two competing estimators of the population mean are: 2 2 2 2 3 2 1 ) 4 4 ( 9 1 )] X ( Var ) X ( Var ) X ( Var [ 9 1 ) X ( Var σ = σ + σ + σ = + + = 2 2 2 2 3 2 1 W 36 24 ) 4 16 4 ( 36 1 )] X ( Var ) X ( Var 16 ) X ( Var [ 36 1 ) X ( Var σ = σ + σ + σ = + + = The results now show: ) X ( Var ) X ( Var W < When the random sample have distributions with unequal population variances then the weighted average W X is more efficient than the sample mean X as an estimator of the population mean.
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Econ 325 – Chapter 7 9 Chapter 7.2 Interval Estimation Let the random sample 1 X , 2 X , . . . , n X be a set of independent and identically distributed random variables with mean μ and variance 2 σ . Consider θ as a population parameter of interest. The true value of this parameter is unknown. A point estimator of θ can be proposed as: ) X , , X , X ( f ˆ n 2 1 K = θ It may also be informative to find random variables low ˆ θ and high ˆ θ such that: 9 . 0 ) ˆ ˆ ( P high low = θ < θ < θ ] ˆ , ˆ [ high low θ θ is called a 90% confidence interval estimator for θ .
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A suggestion is to choose the estimator with minimum...

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