MSE of an estimator

Question

In the book Fundamentals of Statistical Processing, Volume I: Estimation Theory by Steven M. Kay on page 19, it say that the mean square error of an estimator, $\hat{\theta}$, given the true value, $\theta$, is defined as

\begin{align} \text{mse}(\hat{\theta}) &= E[(\hat{\theta} - \theta)^2] \\ &= E \left[ \left[\left( \hat{\theta} - E[\hat{\theta}] \right) + \left( E[\hat{\theta}] - \theta \right) \right]^2 \right]\\ &= \text{var}(\hat{\theta}) + b^2(\theta) \end{align} where $b(\theta) = E[\hat{\theta}] - \theta$ is the bias.

But if I write it out, it doesn't come out to be the same

\begin{align} \text{mse}(\hat{\theta}) &= E[(\hat{\theta} - \theta)^2] \\ &= E \left[ \left[\left( \hat{\theta} - E[\hat{\theta}] \right) + \left( E[\hat{\theta}] - \theta \right) \right]^2 \right]\\ &= E \left[ \left( \hat{\theta} - E[\hat{\theta}] \right)^2 + \left( E[\hat{\theta}] - \theta \right)^2 + 2 \left( \hat{\theta} - E[\hat{\theta}] \right) \left( E[\hat{\theta}] - \theta \right) \right]\\ &= \text{var}(\hat{\theta}) + b^2(\theta) + 2 E \left[\left( \hat{\theta} - E[\hat{\theta}] \right) \left( E[\hat{\theta}] - \theta \right) \right] \end{align}

What am I missing here?

Answer 1

HINT:

$$ E\left[\hat \theta - E\left(\hat \theta \right)\right] = E\left(\hat \theta\right) - E\left(\hat \theta\right) $$

Answer 2

It's important to point out here that Kay is talking about classical estimation, which means that the parameters to be estimated are unknown but deterministic. Hence, the term $E[\hat{\theta}]-\theta$ is deterministic, and, consequently, the last term in your equation becomes

$$\begin{align} E \left[\left( \hat{\theta} - E[\hat{\theta}] \right) \left( E[\hat{\theta}] - \theta \right) \right]&= \left(E[\hat{\theta}]-\theta\right)E\left[\hat{\theta}-E[\hat{\theta}]\right]\\&=\left(E[\hat{\theta}]-\theta\right)\left(E[\hat{\theta}]-E[\hat{\theta}]\right) \\&=0\end{align}$$