Cramer-Rao lower bound

Suppose $X_1,\dots ,X_n\sim P_{\theta }$ and let $\hat{\theta }=\hat{\theta }(X)$ be an unbiased estimator of $\theta$. Then

$$\mathbb{V}(\hat{\theta }(X))\ge \frac{1}{I_n(\theta )}=\frac{1}{n{I}_1(\theta )},$$

where $I(\theta )$ is the Fisher information. Note this shows that the MLE is (asymptotically) optimal, since it achieves this lower bound and is (asymptotically) unbiased.

The multivariate version of the bound is, as you might guess,

$$\mathbb{V}(\hat{\theta }(X))⪰I_n^{-1}(\theta )=\frac{1}{n}{I}_1^{-1}(\theta ).$$

Estimators achieving the Cramer-Rao lower bound are living their best life, you might say.