Cramer-Rao Lower Bound (CRLB)

def. Cramer-Rao Lower Bound. Given unbaised estimator $\hat{\theta }$,

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

thm. There is no other unbiased estimator that has lower preceision than the CRLB.

• Thus CRLB is a statement about the bound on the best possible precision [=efficiency] we can get for unbiased estimators.
• If an estimator reaches the CRLB, it is deemed most efficient.

thm. Cramer-Rao Lower Bount for Biased Estimators. If $\hat{\theta }$ has a

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