Question

Statistics help, please.
On lecture, I have \(MSE(\hat\theta)=E(\hat\theta -\theta)^2=Var(\hat\theta)+(B_{\theta}(\hat\theta))^2\)
I don't get it. Please, explain me

Answer 1

@Zarkon
\((\hat\theta -\theta)^2=\hat\theta^2-2\hat\theta\theta +\theta^2\)
Hence expectation of it is
\(=E(\hat\theta^2)-2E(\hat\theta\theta)+E(\theta ^2)\)

\(E(\hat\theta^2)-(E(\hat\theta))^2+(E(\hat\theta))^2-(2E(\hat\theta\theta)+E(\theta ^2)\)
the first two terms gives \(Var(\hat\theta)\)
but the last three terms = \((B_\theta(\hat\theta))^2\) if \(\hat\theta \) and \(\theta\) independent. Am I right? but they are not independent, right?