Cramer-Rao Lower Bound (CRLB)

thm. Cramer-Rao lower bound. Given estimator $\hat{\theta }$,

$$\begin{gathered} \text{If:}E[\hat{\theta }]=g(\theta ) \\ \Rightarrow \text{Var}(\hat{\theta })\ge \frac{g^{\prime }(\theta )}{\mathcal{I}(\theta )} \end{gathered}$$

If estimator $\hat{\theta }$ is unbiased:

$$\text{Var}(\hat{\theta })\ge \frac{1}{\mathcal{I}(\theta )}$$

⇒ CRLB is a statement about the bound on the best possible precision [=efficiency] we can get.

→ If an estimator reaches the CRLB, it is deemed most efficient.