Hodges' estimator
In
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]
Although Hodges discovered the estimator he never published it; the first publication was in the doctoral thesis of Lucien Le Cam.[5]
Construction
Suppose is a "common" estimator for some parameter : it is consistent, and converges to some asymptotic distribution (usually this is a normal distribution with mean zero and variance which may depend on ) at the -rate:
Then the Hodges' estimator is defined as[6]
This estimator is equal to everywhere except on the small interval , where it is equal to zero. It is not difficult to see that this estimator is consistent for , and its asymptotic distribution is[7]
for any . Thus this estimator has the same asymptotic distribution as for all , whereas for 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 .
It is not true that the Hodges estimator is equivalent to the sample mean, but much better when the true mean is 0. The correct interpretation is that, for finite , the truncation can lead to worse square error than the sample mean estimator for close to 0, as is shown in the example in the following section.[8]
Le Cam shows that this behaviour is typical: superefficiency at the point θ implies the existence of a sequence such that is strictly larger than the
In general, superefficiency may only be attained on a subset of Lebesgue measure zero of the parameter space .[10]
Example
Suppose x1, ..., xn is an
The
See also
Notes
- ^ Vaart (1998, p. 109)
- ^ Kale (1985)
- ^ Bickel (1998, p. 21)
- ^ Vaart (1998, p. 116)
- ^ Le Cam, Lucien M.; University of California, Berkeley. (1953). On some asymptotic properties of maximum likelihood estimates and related Bayes' estimates. University of California publications in statistics; v. 1, no. 11. Berkeley: University of California press.
- ^ Stoica & Ottersten (1996, p. 135)
- ^ Vaart (1998, p. 109)
- ^ Vaart AW van der. Asymptotic Statistics. Cambridge University Press; 1998.
- ^ van der Vaart, A. W., & Wellner, J. A. (1996). Weak Convergence and Empirical Processes. In Springer Series in Statistics. Springer New York. https://doi.org/10.1007/978-1-4757-2545-2
- ^ Vaart AW van der. Asymptotic Statistics. Cambridge University Press; 1998.
- ^ Vaart (1998, p. 110)
References
- Bickel, Peter J.; Klaassen, Chris A.J.; Ritov, Ya’acov; Wellner, Jon A. (1998). Efficient and adaptive estimation for semiparametric models. Springer: New York. ISBN 0-387-98473-9.
- Kale, B.K. (1985). "A note on the super efficient estimator". Journal of Statistical Planning and Inference. 12: 259–263. .
- Stoica, P.; Ottersten, B. (1996). "The evil of superefficiency". Signal Processing. 55: 133–136. .
- Vaart, A. W. van der (1998). Asymptotic statistics. Cambridge University Press. ISBN 978-0-521-78450-4.