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

question 56

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 $\mu$ Then the mean square error (MSE) is defined as follows: $\operatorname { MSE } ( \hat { \mu } ) = E ( \hat { \mu } - \mu ) ^ { 2 } \text {. Prove that } \operatorname { MSE } ( \hat { \mu } ) = \operatorname { bias } ^ { 2 } + \operatorname { var } ( \hat { \mu } ) \text {. }$
(Hint: subtract and add $\left. E ( \hat { \mu } ) \text { in } E ( \hat { \mu } - \mu ) ^ { 2 } . \right)$