# Bias of an Estimator

The quantity ${\displaystyle E_{\theta }(T)-\psi (\theta )}$ is called the bias of estimator T with respect to ${\displaystyle \psi (\theta )}$

If

${\displaystyle E_{\theta }(T)-\psi (\theta )=0}$

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

then the estimator T is an unbiased estimator of ${\displaystyle \psi (\theta )}$.

If T is unbiased,

• ${\displaystyle MSE_{\theta }(T)=var_{\theta }(T)}$
• ${\displaystyle var_{\hat {\theta }}(T)\rightarrow {\sqrt {var_{\hat {\theta }}(T)}}={sd}_{\hat {\theta }}(T)}$