(Advanced)Unbiasedness and small variance are desirable properties of estimators.However,you can imagine situations where a trade-off exists between the two: one estimator may be have a small bias but a much smaller variance than another,unbiased estimator.The concept of "mean square error" estimator combines the two concepts.Let be an estimator of μ.Then the mean square error (MSE)is defined as follows: MSE( )= E( - μ)2.Prove that MSE( )= bias2 + var( ).(Hint: subtract and add in E( )in E( - μ)2. )

Question

(Advanced)Unbiasedness and small variance are desirable properties of estimators.However,you can imagine situations where a trade-off exists between the two: one estimator may be have a small bias but a much smaller variance than another,unbiased estimator.The concept of "mean square error" estimator combines the two concepts.Let  be an estimator of μ.Then the mean square error (MSE)is defined as follows: MSE(  )= E(  - μ)2.Prove that MSE(  )= bias2 + var(  ).(Hint: subtract and add in E(  )in E(  - μ)2. )

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The Answer of (Advanced)Unbiasedness and small variance are desirable properties of estimators.However,you can imagine situations where a trade-off exists between the two: one estimator may be have a small bias but a much smaller variance than another,unbiased estimator.The concept of "mean square error" estimator combines the two concepts.Let  be an estimator of μ.Then the mean square error (MSE)is defined as follows: MSE(  )= E(  - μ)2.Prove that MSE(  )= bias2 + var(  ).(Hint: subtract and add in E(  )in E(  - μ)2. )

(Advanced)Unbiasedness and Small Variance Are Desirable Properties of Estimators