5 chapter 8 examplecontinuedfortherandomsample x 1 x 2

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Unformatted text preview: 2 ) = θ ˆ ˆ θ1 is said to be more efficient than θ 2 if: ˆ ˆ Var (θ1 ) < Var (θ 2 ) The relative efficiency of one estimator with respect to another is the variance ratio: ˆ Var(θ 2 ) ˆ Var(θ1 ) ˆ If θ is an unbiased estimator of θ , and no other unbiased ˆ ˆ estimator has smaller variance than θ , then θ is said to be the most efficient or minimum variance unbiased estimator of θ . 5 Chapter 8 Example Continued: For the random sample X 1 , X 2 , X 3 , introduced above, assume the population variance is σ and also assume independence. 2 That is, Var(X 1 ) = Var(X 2 ) = Var(X 3 ) = σ 2 and Cov(X 1 , X 2 ) = Cov(X 1 , X 3 ) = Cov(X 2 , X 3 ) = 0 Two unbiased estimators for the population mean were proposed as: 1 X = (X 1 + X 2 + X 3 ) 3 and 1 X W = (X 1 + 4 X 2 + X 3 ) 6 The variance of the first estimator is: 1 Var( X ) = [Var(X 1 ) + Var(X 2 ) + Var(X 3 )] 9 1 = (3 σ 2 ) = σ 3 9 2 The variance of the second estimator is: Var( X W ) = 6 1 [Var(X 1 ) + 16 Var(X 2 ) + Var(X 3 )] 36 2 1 = (18 σ 2 ) = σ 2 36 Chapter 8 It can be seen that: Var ( X ) < Var ( X W ) Therefore, X is more efficient than X W . The relative efficiency is: 2 Var( X W ) σ 2 3 = 2 = = 1.5 Var( X ) σ3 2 Note: A reporting style for the measure of relative efficiency is to place the higher variance in the numerator. 7 Chapter 8 As a variation, suppose that the random sample X 1 , X 2 , X 3 have probability distributions with identical population mean μ but unequal population variances. Assume: Var(X 1 ) = 4 σ 2 Var(X 2 ) = σ 2 Var(X 3 ) = 4 σ 2 and The revised variances for the two competing estimators of the population mean are: 1 Var( X ) = [Var(X 1 ) + Var(X 2 ) + Var(X 3 )] 9 1 = (4 σ 2 + σ 2 + 4 σ 2 ) 9 = σ2 1 [Var(X 1 ) + 16 Var(X 2 ) + Var(X 3 )] 36 1 = (4 σ 2 + 16 σ 2 + 4 σ 2 ) 36 Var( X W ) = = 24 2 σ 36 The results now show: Var ( X W ) < Var ( X ) When the random sample have distributions with population W variances unequal then the weighted average X is more efficient than the sample mean X as an estimator of the population mean. 8 Chapter 8 Chapter 8.2 Interval Estimation Let the random sample X 1 , X 2 , . . . , X n be a set of independent and identically distributed random variables with mean μ and variance σ . 2 Consider θ as a population parameter of interest. The true value of this parameter is unknown. ˆ A point estimator of θ can be proposed as: θ = f( X 1 , X 2 , K , X n ) It may also be informative to find random variables ˆ ˆ θ low and θ high such that: ˆ ˆ P(θ low < θ < θ high ) = 0.9 ˆ ˆ [θ low , θ high ] is called a 90% confidence interval estimator for θ . ˆ ˆ In general, find ran...
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