Mean Squared Error

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

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

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

for every $\theta \in \Omega$$\theta \in \Omega$,

then

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

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

Decomposition of MSE

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

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

PROOF

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

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