Question
Show that the mean square estimation error satisfies $E\left[(\hat{\Theta}-\theta)^2\right]=\operatorname{VAR}[\hat{\Theta}]+B(\hat{\Theta})^2$.
Step 1
This expression represents the expected value of the squared difference between the estimator \( \hat{\Theta} \) and the true parameter \( \theta \). Show more…
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Using the identity $$(\hat{\theta}-\theta)=[\hat{\theta}-E(\hat{\theta})]+[E(\hat{\theta})-\theta]=[\hat{\theta}-E(\hat{\theta})]+B(\hat{\theta})$$ show that $$ \operatorname{MSE}(\hat{\theta})=E\left[(\hat{\theta}-\theta)^{2}\right]=V(\hat{\theta})+(B(\hat{\theta}))^{2} $$
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Suppose that $\hat{\theta}$ is an estimator for a parameter $\theta$ and $E(\hat{\theta})=a \theta+b$ for some nonzero constants $a$ and $b$ a. In terms of $a, b,$ and $\theta$, what is $B(\hat{\theta}) ?$ b. Find a function of $\hat{\theta}-\operatorname{say}, \hat{\theta}^{*}-$ that is an unbiased estimator for $\theta$
a. If $\hat{\theta}$ is an unbiased estimator for $\theta$, what is $B(\hat{\theta}) ?$ b. If $B(\hat{\theta})=5,$ what is $E(\hat{\theta}) ?$
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