Lecture_23

Lecture_23 - 1 MEASURES OF THE ESTIMATORS Lecture 23...

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Unformatted text preview: 1 MEASURES OF THE ESTIMATORS Lecture 23 ORIE3500/5500 Summer2009 Chen Class Today • Measures of the Estimators • Maximum Likelihood Estimates 1 Measures of the Estimators A point estimator of a parameter θ of the distribution of the observations X 1 ,...,X n is any suitable statistic ˆ θ = ˆ θ ( X 1 ,...,X n ) and a value obtained by evaluating the function for the observed values of the random variables is a point estimate . We have already discussed that an estimator ˆ θ is unbiased for a parameter θ if E £ ˆ θ ( X 1 ,...,X n ) / = θ, otherwise ˆ θ is a biased estimator of θ . For a biased estimator ˆ θ of the para- meter θ , the bias of ˆ θ is defined as bias ( ˆ θ ) = E £ ˆ θ ( X 1 ,...,X n ) /- θ. So bias is a measure of how good the estimator is. Another such measure is consistency. An estimator ˆ θ is consistent for θ if, for any ² > 0, P ( | ˆ θ ( X 1 ,...,X n )- θ | > ² ) → ,n → ∞ . We also say that ˆ θ converges to θ in probability. This means that for large n we can expect that ˆ θ ( X 1 ,...,X n ) is close to θ for large values of n . The mean squared error is another measure for the precision of an esti- mator. The name is self explanatory MSE ( ˆ θ ) = E £ ( ˆ θ ( X 1 ,...,X n )- θ ) 2 / Facts: For any estimator ˆ θ of θ , MSE( ˆ θ ) = Bias( ˆ θ ) 2 + Var( ˆ θ ) 1 2 MAXIMUM LIKELIHOOD ESTIMATES Proof. MSE( ˆ θ ) = E ( ˆ θ- θ ) 2 = E [ { ˆ θ- E ( ˆ θ ) } + { E ( ˆ θ )- θ } ] 2 = E { ˆ θ- E ( ˆ θ ) } 2 + { E ( ˆ θ )- θ } 2 + 2 { E ( ˆ θ )- θ } E { ˆ θ- E ( ˆ θ ) } = Var( ˆ θ ) + Bias( ˆ θ ) 2 where the last step follows since 2 { E ( ˆ θ )- θ } E { ˆ θ- E ( ˆ θ ) } = 2 { E ( ˆ θ )- θ }{ E ( ˆ θ )- E ( ˆ θ ) } = 0 Intuitively, comparing the MSE is the most natural way to compare two estimators ˆ θ 1 and ˆ θ 2 , i.e....
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This note was uploaded on 09/22/2009 for the course ORIE 3500 taught by Professor Weber during the Summer '08 term at Cornell.

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Lecture_23 - 1 MEASURES OF THE ESTIMATORS Lecture 23...

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