Question about proof $\mathbb{E}[(\hat{\theta} - \theta)^2]=\mathbb{V}(\hat{\theta}) + \mathbb{E}(\hat{\theta})- \theta$

Question

Let's say $\theta$ is a poblational quantity (a moment of a random variable, for example) and $\hat{\theta} = \hat{\theta}(X_1, X_2, \ldots ,X_n)$ is an estimator of this quantity. I want to prove that $$\mathbb{E}[(\hat{\theta} - \theta)^2]=\mathbb{V}(\hat{\theta}) + \mathbb{E}(\hat{\theta})- \theta$$ I have seen a prove in a book but I don't understand the following step: $$\mathbb{E}\left[\left(\hat{\theta}-\mathbb{E}(\hat{\theta})\right)\left(\mathbb{E}(\hat{\theta}) - \theta\right)\right] = \left(\mathbb{E}(\hat{\theta}) - \theta\right) \cdot \left(\mathbb{E}(\hat{\theta})-\mathbb{E}(\hat{\theta})\right)$$ First of all, I can't see why you can't take $\left(\mathbb{E}(\hat{\theta}) - \theta\right)$ outside of the expectation operator. Second of all, I don't see why $\mathbb{E}\left[\hat{\theta}-\mathbb{E}(\hat{\theta})\right] =\left(\mathbb{E}(\hat{\theta})-\mathbb{E}(\hat{\theta})\right)$
Can someone please help me understand this?

Answer 1

You can take the $\left( \mathbb{E}(\hat{\theta}) - \theta \right)$ outside of the expectation because it's not random. The only random variable involved is the estimator $\hat{\theta}$; the quantities $\mathbb{E}(\hat{\theta})$ and $\theta$ are both just numbers.

The equality $\mathbb{E} \left[ \hat{\theta} - \mathbb{E}(\hat{\theta}) \right] = \left( \mathbb{E}(\hat{\theta}) - \mathbb{E}(\hat{\theta}) \right)$ comes from linearity of expectation and again the fact that $\mathbb{E}(\hat{\theta})$ is just a number (not random): $$ \mathbb{E} \left[ \hat{\theta} - \mathbb{E}(\hat{\theta}) \right] = \mathbb{E}(\hat{\theta}) - \mathbb{E} \left( \mathbb{E}(\hat{\theta}) \right) = \mathbb{E}(\hat{\theta}) - \mathbb{E}(\hat{\theta}) $$