NotesNov15 - Measuring the precision of estimators The most...

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Measuring the precision of estimators The most commong measure of precision: the Mean Squared Error (MSE). The MSE of an estimator ˆ θ of θ is MSE ( ˆ θ ) = E ( ˆ θ - θ ) 2 = E ( ˆ θ ( X 1 , . . . , X n ) - θ ) 2 .
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For an unbiased estimator ˆ θ MSE ( ˆ θ ) = Var( ˆ θ ) . For a biased estimator ˆ θ , MSE ( ˆ θ ) = Var( ˆ θ ) + bias( ˆ θ ) 2 . The MSE of an estimator is equal to its variance plus its squared bias .
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Example : Sample mean For a sample from any distribution, the sample mean ¯ X n is an unbiased estimator of the true mean μ . The mean square error: MSE ( ¯ X n ) = Var( ¯ X n ) = 1 n σ 2 , where σ 2 is the population variance. By the law of large numbers ¯ X n μ as n , so the sample mean is also a consistent estimator of the true mean.
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Example : Sample variance For a sample from any distribution the sample variance S 2 is an unbiased estimator of the true variance σ 2 . Furthermore, S 2 σ 2 as the sample size increases.
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