Bias of an Estimator

Definition

Let $X_1,X_2,\dots ,X_n$ be a Random Sample from $f(\mathbf{x}|\mathbf{\theta })$ where $\mathbf{\theta }\in \Omega$ is a parameter, and $\hat{\mathbf{\theta }}=T(X_1,\dots ,X_n)$ be a Statistic. $\mathrm{Bias}(\hat{\mathbf{\theta }})=E((\hat{\mathbf{\theta }}))-\mathbf{\theta }$

$\forall \mathbf{\theta }\in \Omega ,E(\hat{\mathbf{\theta }})=\mathbf{\theta }\Rightarrow \hat{\mathbf{\theta }}$ is unbiased estimator An estimator is unbiased if its bias is equal to zero for all values of parameter $\mathbf{\theta }$.