# Mean Squared Error

The Mean Square Error (MSE) of an estimator ${\displaystyle T}$ of ${\displaystyle \psi (\theta )}$ is ${\displaystyle MSE_{\theta }(T)=E_{\theta }}${${\displaystyle [T-\psi (\theta )]^{2}}$}

• Two estimates of ${\displaystyle \psi (\theta )}$: ${\displaystyle T_{1}}$ and ${\displaystyle T_{1}}$

If ${\displaystyle MSE_{\theta }(T_{1})\leq {MSE}_{\theta }(T_{2})}$

for every ${\displaystyle \theta \in \Omega }$,

then

${\displaystyle T_{1}}$ is better than ${\displaystyle T_{2}}$.

• ${\displaystyle MSE_{\hat {\theta }}(T)}$

### Decomposition of MSE

${\displaystyle MSE_{\theta }(T)=var_{\theta }(T)+[E_{\theta }(T)-\psi (\theta )]^{2}}$

PROOF

{\displaystyle {\begin{aligned}E_{\theta }{\Bigl (}[T-\psi (\theta )]^{2}{\Bigr )}&=E_{\theta }{\Bigl (}[T-E_{\theta }(T)+E_{\theta }(T)-\psi (\theta )]^{2}{\Bigr )}\\&=E_{\theta }{\Bigl (}[T-E_{\theta }(T)]^{2}{\Bigr )}+E_{\theta }{\Bigl (}[E_{\theta }(T)-\psi (\theta )]^{2}{\Bigr )}+2E_{\theta }{\Bigl (}[T-E_{\theta }(T)][E_{\theta }(T)-\psi (\theta )]{\Bigr )}\\&={\text{var}}_{\theta }(T)+[E_{\theta }(T)-\psi (\theta )]^{2}+2[E_{\theta }(T)-\psi (\theta )]\overbrace {E_{\theta }{\Bigl (}[T-E_{\theta }(T)]{\Bigr )}} ^{0}\\&={\text{var}}_{\theta }(T)+[E_{\theta }(T)-\psi (\theta )]^{2}\end{aligned}}}