post10 - Quality of Estimators Example f (x) = 1/2, Mean...

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Quality of Estimators Example f ( x )= 1 / 2 - 100 <x<θ - 99 , 1 / 2 +99 + 100 , 0 otherwise Let u ( x x . Then, it is an unbiased estimator of θ . But will it ever be close to θ ? Unbiasedness is not enough. 1 Mean Squared Error (MSE) The mean square error(MSE) of an estimator ˆ θ = u ( X 1 , ··· ,X n )fo r θ is: MSE( ˆ θ E (( ˆ θ - θ ) 2 ) =var( ˆ θ )+(Bias( ˆ θ )) 2 A small MSE is a good property for an estima- tor to have. 2 Example : X 1 , n : iid N ( μ, σ 2 ) Consider two estimator of σ 2 : S 2 n = 1 n X ( X i - ¯ X ) 2 S * 2 n = 1 n - 1 X ( X i - ¯ X ) 2 . How do these compare in terms of MSE? 3 MSE and Risk MSE is a specific example of a more general concept called risk. Risk is defined as the ex- pected loss: R ( θ, u ( X 1 , n )) = E (Loss( θ, u ( X 1 , n )) Loss function measures the distance between the estimate and the parameter. (e.g. Loss( θ, u ( u - θ ) 2 ) Can we find the estimator with the smallest risk? With the squared error loss, MSE and risk are the same. Can we find the estimator with the
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post10 - Quality of Estimators Example f (x) = 1/2, Mean...

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