Then θ ˆ is said to be the most efficient or

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, 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 . In general, find random variables low ˆ θ and high ˆ θ such that: α - = θ < θ < θ 1 ) ˆ ˆ ( P high low Greek letter alpha α - 1 is called the confidence level . ] ˆ , ˆ [ high low θ θ gives a ) 1 ( α - 100 % confidence interval estimator for θ . Econ 325 – Chapter 7 10 xrhombus Interval Estimation for the Population Mean Results established in previous lecture notes are first reviewed. A point estimator for the population mean μ is: = = n 1 i i X n 1 X X has a sampling distribution with the properties: μ = ) X ( E ( X is an unbiased estimator of μ ) and n ) X ( Var 2 σ = The standard error of X is: n ) X ( se σ = To proceed further, assume that X follows a normal distribution. This assumption is reasonable since: If 1 X , 2 X , . . . , n X follow a normal distribution then a result is that X also has a normal distribution.
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