In Section 9.3, we emphasized the notion of most efficient estimator by comparing the variance of two unbiased estimators 1 and 2. However,

In Section 9.3, we emphasized the notion of

“most efficient estimator” by comparing the variance of two unbiased estimators Θˆ 1 and Θˆ 2. However, this does not take into account bias in case one or both estimators are not unbiased. Consider the quantity MSE = E(Θˆ − θ), where MSE denotes mean squared error. The MSE is often used to compare two estimators Θˆ 1 and

Θˆ 2 of θ when either or both is unbiased because (i) it is intuitively reasonable and (ii) it accounts for bias.

Show that MSE can be written MSE = E[Θˆ − E(Θ)] ˆ 2 + [E(Θˆ − θ)]2

= Var(Θ) + [Bias( ˆ Θ)] ˆ 2

.