# Hodges' estimator

In statistics, **Hodges’ estimator**^{[1]} (or the **Hodges–Le Cam estimator**^{[2]}), named for Joseph Hodges, is a famous^{[3]} 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.^{[4]}

## Construction

Suppose 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 Template:Sqrt-rate:

Then the **Hodges’ estimator** is defined as ^{[5]}

This estimator is equal to everywhere except on the small interval [−*n*^{−1/4}, *n*^{−1/4}], where it is equal to zero. It is not difficult to see that this estimator is consistent for *θ*, and its asymptotic distribution is ^{[6]}

for any *α* ∈ **R**. Thus this estimator has the same asymptotic distribution as for all *θ* ≠ 0, whereas for *θ* = 0 the rate of convergence becomes arbitrarily fast. This estimator is *superefficient*, as it surpasses the asymptotic behavior of the efficient estimator at least at one point *θ* = 0. In general, superefficiency may only be attained on a subset of measure zero of the parameter space Θ.

## Example

Suppose *x*_{1}, …, *x _{n}* is an iid sample from normal distribution

*N*(

*θ*, 1) with unknown mean but known variance. Then the common estimator for the population mean

*θ*is the arithmetic mean of all observations: . The corresponding Hodges’ estimator will be , where

**1**{…} denotes the indicator function.

The mean square error (scaled by *n*) associated with the regular estimator *x* is constant and equal to 1 for all *θ*’s. At the same time the mean square error of the Hodges’ estimator behaves erratically in the vicinity of zero, and even becomes unbounded as *n* → ∞. This demonstrates that the Hodges’ estimator is not regular, and its asymptotic properties are not adequately described by limits of the form (*θ* fixed, *n* → ∞).

## See also

## Notes

### References

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