Unbiased Estimator

Definition

Let

• $X_1,\dots ,X_n$ : Random samples, with
  • $\Omega$ : Parameter space
  • pdf $f(x;\theta ),\theta \in \Omega$
• $T$ : Statistic of $X_1,\dots ,X_n$

If $E(T)=\theta ,\forall \theta \in \Omega$

Then we say $T$ is an unbiased estimator of $\theta$

Remark

More formally, $T$ is called a point estimator of $\theta$. We call its realization $t$ an estimate of $\theta$.

Related theorems

Obtain

• Rao-Blackwell : $E(Y_2|Y_1)=\phi (Y_1)$, $Y_2$ sufficient, $Y_1$ unbiased

Use

• Rao-Cramer Lower Bound for Unbiased Estimators