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

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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.)

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    $\begingroup$ 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? $\endgroup$ – StubbornAtom Jan 25 at 9:55
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    $\begingroup$ @StubbornAtom: The Cramèr-Rao bound lemma applies under regularity conditions that are not satisfied by the Uniform distribution, there is thus no surprise whatsoever that it is not a valid bound on the unbiased estimator in the Uniform case. The OP was apparently asking for cases as such. $\endgroup$ – Xi'an Feb 11 at 6:13

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