{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

chap07PRN econ 325

# A suggestion is to choose the estimator with minimum

This preview shows pages 3–6. Sign up to view the full content.

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 = + + =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page3 / 14

A suggestion is to choose the estimator with minimum...

This preview shows document pages 3 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online