(Advanced)Unbiasedness and Small Variance Are Desirable Properties of Estimators $\hat { \mu }$

question 43

Essay

(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 $\hat { \mu }$ be an estimator of μ. Then the mean square error (MSE)is defined as follows: MSE( $\hat { \mu }$ )= E( $\hat { \mu }$ - μ)². Prove that MSE( $\hat { \mu }$ )= bias² + var( $\hat { \mu }$ ). (Hint: subtract and add in E( $\hat { \mu }$ )in E( $\hat { \mu }$ - μ)².)

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