Is there any parameter space of Cramér–Rao_bound

Question

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?

Answer 1

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:

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.

Passage to a limit should always be the last operation, not the first.