Hodges’ estimator

In statistics, Hodges’ estimator (or the Hodges–Le Cam estimator), named for Joseph Hodges, is a famous counter example of an estimator which is "superefficient", i.e. it attains smaller asymptotic variance than regular efficient estimators. The existence of such a counterexample is the reason for the introduction of the notion of regular estimators.

Hodges’ estimator improves upon a regular estimator at a single point. In general, any superefficient estimator may surpass a regular estimator at most on a set of Lebesgue measure zero.

Suppose ${\displaystyle \scriptstyle {\hat {\theta }}_{n}}$ is a "common" estimator for some parameter θ: it is consistent, and converges to some asymptotic distribution L_θ (usually this is a normal distribution with mean zero and variance which may depend on θ) at the √n-rate:

Then the Hodges’ estimator ${\displaystyle \scriptstyle {\hat {\theta }}_{n}^{H}}$ is defined as

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