Hodges' estimator

Construction

Suppose θ ^ 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 θ ^ n H is defined as

This estimator is equal to θ ^ n 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

for any α ∈ R. Thus this estimator has the same asymptotic distribution as θ ^ n 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 θ ^ n 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 x₁, …, 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: x ¯ . The corresponding Hodges’ estimator will be θ ^ n H = x ¯ ⋅ 1 { | x ¯ | ≥ n − 1 / 4 } , 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 θ ^ n H 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 → ∞).