It is known that Cramér–Rao\_bound is the lower bound of variance of a parameter. A useful link is <https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound> There is also a term called 'superefficient', mentioned in the link: <https://en.wikipedia.org/wiki/Hodges%27_estimator> which claims that: "In general, any superefficient estimator may surpass a regular estimator at most on a set of Lebesgue measure zero". My question is how "Lebesgue measure zero" is associated with the parameters discussed in the Cramér–Rao\_bound? Is there a limitation where Cramér–Rao\_bound is applicable?



Is there any parameter space of Cramér–Rao_bound
The basic form of the Crammer-Rao bound is: if $\hat \theta$ is **unbiased** (or at least locally unbiased around $\theta\_0$) then:


$$ var(\hat \theta | \theta\_0) \geq I\_F^{-1}(\theta\_0) $$


The general form is: define the function $e(\theta) = E(\hat \theta | \theta)$. Then:


$$ var(\hat \theta | \theta\_0) \geq e'(\theta\_0) I\_F^{-1}(\theta\_0) e'(\theta\_0) $$


(If I haven't messed up, this formula should work for vector valued $\theta$: that's why I separated the derivatives). You can see how the general formula reduces to the basic formula when $e' = 1$


Any super-efficient guy is "cheating" the Cramer-Rao bound by being biased. While they break the basic Crao bound, they still respect the general formula for biased estimators. If you consider the Hodges, for every finite $n$, $\hat \theta\_n^H$ is biased everywhere, and actually $e'(\theta=0)=0$ so that it's not surprising that they can reach an arbitrarily low variance there.


I think this super-efficient business looks fishy. The hodge estimator appears surpisingly powerful because we went to the limit $n \rightarrow \infty$ and then considered his properties. That's stupid: at $n=\infty$ most non-absurd estimators are exact. Nobody can be better than anybody else. When we looked at the property of the hodge estimator for a finite n, the "paradox" disappeared. To quote Jaynes:



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> Not only in probability theory, but in all mathematics, it is the careless use of infinite sets, and of infinite and infinitesimal quantities, that generates most paradoxes.
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> Passage to a limit should always be the last operation, not the first.
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