# Cramèr-Rao lower bound on the variance of an unbiased estimator

When is the variance of an unbiased estimator lower than the Cramèr-Rao lower bound ??

For distributions that do not satisfy the conditions for the Cramèr-Rao lemma to apply, e.g. for these which support depends on the parameter, there is no reason for the Cramèr-Rao inequality to hold. See for instance this example from Casella & Berger (Example 7.3.13), when $$X_1,\ldots,X_n\sim\mathcal{U}(0,\theta)$$ the estimator $$\frac{n}{n-1}X_{(n)}$$ is unbiased and with variance $$\theta^2/n(n+2)$$ while the Cramèr-Rao lower bound is $$\theta^2/n$$. (See this entry on X validated about the Fisher information for Uniforms, which some argue does not exist. I tend to agree.)
• Indeed the Fisher information $I(\theta)$ does not exist here (we can arrive at different expressions for $I(\theta)$ using different formulae). But does the CR inequality really apply in these cases since the regularity conditions are not satisfied? – StubbornAtom Jan 25 at 9:55