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In [[statistics]], '''Hodges’ estimator'''<ref>{{harvtxt|Vaart|1998|p=109}}</ref> (or the '''Hodges–Le Cam estimator'''<ref>{{harvtxt|Kale|1985}}</ref>), named for [[Joseph Lawson Hodges, Jr.|Joseph Hodges]], is a famous<ref>{{harvtxt|Bickel|1998|p=21}}</ref> counter example of an [[estimator]] which is "superefficient", i.e. it attains smaller asymptotic variance than regular [[efficient estimator]]s. The existence of such a counterexample is the reason for the introduction of the notion of [[regular estimator]]s.
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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.<ref>{{harvtxt|Vaart|1998|p=116}}</ref>
 
== Construction ==
Suppose <math style="vertical-align:-.1em">\scriptstyle\hat\theta_n</math> is a "common" estimator for some parameter ''θ'': it is [[consistent estimator|consistent]], and converges to some [[asymptotic distribution]] ''L<sub>θ</sub>'' (usually this is a [[normal distribution]] with mean zero and variance which may depend on ''θ'') at the {{nowrap|{{sqrt|''n''}}-rate}}:
: <math>
    \sqrt{n}(\hat\theta_n - \theta)\ \xrightarrow{d}\ L_\theta\ .
  </math>
 
Then the '''Hodges’ estimator''' <math style="vertical-align:-.2em">\scriptstyle\hat\theta^H_n</math> is defined as <ref>{{harvtxt|Stoica|Ottersten|1996|p=135}}</ref>
: <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 style="vertical-align:-.1em">\scriptstyle\hat\theta_n</math> everywhere except on the small interval {{nowrap|[−''n''<sup>−1/4</sup>, ''n''<sup>−1/4</sup>]}}, where it is equal to zero. It is not difficult to see that this estimator is [[consistent estimator|consistent]] for ''θ'', and its [[asymptotic distribution]] is <ref>{{harvtxt|Vaart|1998|p=109}}</ref>
: <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 ''α'' ∈ '''R'''. Thus this estimator has the same asymptotic distribution as <math style="vertical-align:-.1em">\scriptstyle\hat\theta_n</math> for all {{nowrap|''θ'' ≠ 0}}, whereas for {{nowrap|''θ'' {{=}} 0}} the rate of convergence becomes arbitrarily fast. This estimator is ''superefficient'', as it surpasses the asymptotic behavior of the efficient estimator <math style="vertical-align:-.1em">\scriptstyle\hat\theta_n</math> at least at one point {{nowrap|''θ'' {{=}} 0}}. In general, superefficiency may only be attained on a subset of measure zero of the parameter space Θ.
 
== Example ==
[[Image:Hodges estimator risk function.svg|right|thumb|The [[mean square error]] (times ''n'') of Hodges’ estimator. Blue curve corresponds to {{nowrap|''n'' {{=}} 5}}, purple to {{nowrap|''n'' {{=}} 50}}, and olive to {{nowrap|''n'' {{=}} 500}}.<ref>{{harvtxt|Vaart|1998|p=110}}</ref>]]
 
Suppose ''x''<sub>1</sub>, …, ''x<sub>n</sub>'' is an [[iid]] sample from normal distribution {{nowrap|''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: <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 <span style="text-decoration:overline">''x''</span> is constant and equal to 1 for all ''θ''’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 {{nowrap|''n'' → ∞}}. This demonstrates that the Hodges’ estimator is not [[regular estimator|regular]], and its asymptotic properties are not adequately described by limits of the form (''θ'' fixed, {{nowrap|''n'' → ∞}}).
 
== See also ==
* [[James–Stein estimator]]
 
== Notes ==
{{reflist|2}}
 
=== References ===
{{refbegin}}
* {{cite book
  | last1 = Bickel  | first1 = Peter J.
  | last2 = Klaassen | first2 = Chris A.J.
  | last3 = Ritov    | first3 = Ya’acov
  | last4 = Wellner  | first4 = Jon A.
  | year = 1998
  | title = Efficient and adaptive estimation for semiparametric models
  | publisher = Springer: New York
  | isbn = 0-387-98473-9
  | ref = CITEREFBickel1998
  }}
* {{cite journal
  | last = Kale | first = B.K.
  | year = 1985
  | title = A note on the super efficient estimator
  | journal = Journal of Statistical Planning and Inference
  | volume = 12
  | pages = 259–263
  | doi = 10.1016/0378-3758(85)90074-6
  | ref = harv
  }}
* {{cite journal
  | last1 = Stoica    | first1 = P.
  | last2 = Ottersten | first2 = B.
  | year = 1996
  | title = The evil of superefficiency
  | journal = Signal Processing
  | volume = 55
  | pages = 133–136
  | doi = 10.1016/S0165-1684(96)00159-4
  | ref = harv
  }}
* {{cite book
  | last = Vaart | first = A. W. van der
  | year = 1998
  | title = Asymptotic statistics
  | publisher = Cambridge University Press
  | isbn = 978-0-521-78450-4
  | ref = harv
  }}
{{refend}}
 
[[Category:Estimation theory]]

Latest revision as of 01:19, 23 August 2014

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