Английская Википедия:Hodges' estimator
Шаблон:Short descriptionIn statistics, Hodges' estimator[1] (or the Hodges–Le Cam estimator[2]), named for Joseph Hodges, is a famous counterexample of an estimator which is "superefficient",[3] 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]
Although Hodges discovered the estimator he never published it; the first publication was in the doctoral thesis of Lucien Le Cam.[5]
Construction
Suppose <math>\hat{\theta}_n</math> is a "common" estimator for some parameter <math>\theta</math>: it is consistent, and converges to some asymptotic distribution <math>L_\theta</math> (usually this is a normal distribution with mean zero and variance which may depend on <math>\theta</math>) at the <math>\sqrt{n}</math>-rate:
- <math>
\sqrt{n}(\hat\theta_n - \theta)\ \xrightarrow{d}\ L_\theta\ .
</math>
Then the Hodges' estimator <math>\hat{\theta}_n^H</math> is defined as[6]
- <math>
\hat\theta_n^H = \begin{cases}\hat\theta_n, & \text{if } |\hat\theta_n| \geq n^{-1/4}, \text{ and} \\ 0, & \text{if } |\hat\theta_n| < n^{-1/4}.\end{cases}
</math>
This estimator is equal to <math>\hat{\theta}_n</math> everywhere except on the small interval <math>[-n^{-1/4},n^{-1/4}]</math>, where it is equal to zero. It is not difficult to see that this estimator is consistent for <math>\theta</math>, and its asymptotic distribution is[7]
- <math>\begin{align}
& n^\alpha(\hat\theta_n^H - \theta) \ \xrightarrow{d}\ 0, \qquad\text{when } \theta = 0, \\
&\sqrt{n}(\hat\theta_n^H - \theta)\ \xrightarrow{d}\ L_\theta, \quad \text{when } \theta\neq 0,
\end{align}</math>
for any <math>\alpha\in\mathbb{R}</math>. Thus this estimator has the same asymptotic distribution as <math>\hat{\theta}_n</math> for all <math>\theta\neq 0</math>, whereas for <math>\theta=0</math> the rate of convergence becomes arbitrarily fast. This estimator is superefficient, as it surpasses the asymptotic behavior of the efficient estimator <math>\hat{\theta}_n</math> at least at one point <math>\theta=0</math>.
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 <math>n</math>, the truncation can lead to worse square error than the sample mean estimator for <math>E[X]</math> 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 <math>\theta_n \rightarrow \theta</math> such that <math>\lim \inf E \theta_n \ell (\sqrt n (\hat \theta_n - \theta_n ))</math> is strictly larger than the Cramer-Rao bound. For the extreme case where the asymptotic risk at θ is zero, the <math>\liminf</math> is even infinite for a sequence <math>\theta_n \rightarrow \theta</math>.[9]
In general, superefficiency may only be attained on a subset of Lebesgue measure zero of the parameter space <math>\Theta</math>.[10]
Example
Suppose x1, ..., xn is an independent and identically distributed (IID) random sample from normal distribution Шаблон:Nowrap with unknown mean but known variance. Then the common estimator for the population mean θ is the arithmetic mean of all observations: <math style="vertical-align:0">\scriptstyle\bar{x}</math>. The corresponding Hodges' estimator will be <math style="vertical-align:-.35em">\scriptstyle\hat\theta^H_n \;=\; \bar{x}\cdot\mathbf{1}\{|\bar x|\,\geq\,n^{-1/4}\}</math>, 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 θШаблон:'s. At the same time the mean square error of the Hodges' estimator <math style="vertical-align:-.3em">\scriptstyle\hat\theta_n^H</math> behaves erratically in the vicinity of zero, and even becomes unbounded as Шаблон:Nowrap. This demonstrates that the Hodges' estimator is not regular, and its asymptotic properties are not adequately described by limits of the form (θ fixed, Шаблон:Nowrap).
See also
Notes
References
- ↑ Шаблон:Harvtxt
- ↑ Шаблон:Harvtxt
- ↑ Шаблон:Harvtxt
- ↑ Шаблон:Harvtxt
- ↑ Шаблон:Cite book
- ↑ Шаблон:Harvtxt
- ↑ Шаблон:Harvtxt
- ↑ 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.
- ↑ Шаблон:Harvtxt
