# Actuarial Empire

## Stats Review 1: A Proof of the MSE formula

The formal definition for $MSE_{\theta}(\hat{\theta}) = E_{\theta}[(\hat{\theta} – \theta)^2]$. However, it is often more convenient to write $MSE_{\theta}(\hat{\theta}) = Var_\theta(\hat{\theta}) + Bias_\theta(\hat{\theta})^2$. Here’s a proof of this result:
\begin{eqnarray*}
MSE_{\theta}(\hat{\theta}) &=& E_{\theta}\left[\left(\hat{\theta} – \theta\right)^2\right] \\
&=& E_{\theta}\left[ \left ([\hat{\theta} – E_\theta(\hat{\theta})] + [E_\theta(\hat{\theta}) – \theta] \right)^2\right]\\
&=& E\left[ \left( \hat{\theta} – E_\theta(\hat{\theta}) \right)^2 \right] + 2E_\theta \left [ \left(\hat{\theta} – E_\theta(\hat{\theta}) \right)\left(E_\theta(\hat{\theta}) – \theta \right) \right] + E_\theta\left[ E_\theta(\hat{\theta}) – \theta \right]^2 \\
&=& Var_\theta(\hat{\theta}) + 2E_\theta \left[ \left( E_\theta(\hat{\theta})(\hat{\theta} -E_\theta(\hat{\theta})\right) – \theta \left(\hat{\theta} – E_\theta(\hat{\theta}) \right) \right ] + Bias_\theta(\hat{\theta})^2 \\
&=& Var_\theta(\hat{\theta}) + 2\times(0-0) +Bias_\theta(\hat{\theta})^2 \\
&=& Var_\theta(\hat{\theta}) +Bias_\theta(\hat{\theta})^2
\end{eqnarray*}

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