Rao-Cramer Lower Bound

Definition

Let

• $X_1,\dots ,X_n$: Random sample, with pdf $f(x;\theta ),\theta \in \Omega$
• $Y=u(X_1,X_2,\dots ,X_n)$ : Statistic, with Mean $E(Y)=k(\theta )$
• $I(\theta )$ : Fisher information

Assume regularity conditions and additional regularity conditions 1 hold.

Then $\mathrm{Var}(Y)\ge \boxed{\frac{[k^{\prime }(\theta ){]}^2}{nI(\theta )}}$

• We say $[k^{\prime }(\theta ){]}^2/nI(\theta )$ is the Rao-Cramer lower bound of $Y$

Remark

The Rao-Cramér lower bound provides a theoretical minimum variance for unbiased estimators when regularity conditions hold.

If the bound cannot be computed or doesn’t seem to apply, then one or more regularity conditions may have been violated.