Cramér-Rao Lower Bound

Topics: Statistics

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(theorem)

Let $X_1,\dots ,X_n\sim f(x\mid \theta )$, with $\theta \in \Theta$ (i.e. a parameter in a parameter space).

Let $T$ be an estimator of $\theta$. Then:

$$\mathrm{Var}(T)\ge \frac{1}{n\mathbb{E}\left[{\left(\frac{d\ln \left(f(x\mid \theta )\right)}{d\theta }\right)}^2\right]}:=\mathrm{CRLB}(\theta )$$

In other words, the Cramér-Rao lower bound ($\mathrm{CRLB}(\theta )$) provides a lower bound for $T$’s variance.