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Difference between revisions of "Asymptotically-efficient estimator"

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A concept which extends the idea of an efficient estimator to the case of large samples (cf. [[Efficient estimator|Efficient estimator]]). An asymptotically-efficient estimator has not been uniquely defined. Thus, in its classical variant it concerns the asymptotic efficiency of an estimator in a suitably restricted class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013800/a0138001.png" /> of estimators. In fact, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013800/a0138002.png" /> be a [[Consistent estimator|consistent estimator]] of a one-dimensional parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013800/a0138003.png" /> constructed from a random sample of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013800/a0138004.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013800/a0138005.png" /> if the variance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013800/a0138006.png" /> exists, and if it is bounded from below, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013800/a0138007.png" />, by the inverse of the [[Fisher amount of information|Fisher amount of information]] corresponding to one observation. An estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013800/a0138008.png" /> which attains the lower bound just mentioned is asymptotically efficient. Under certain conditions this property is satisfied by the maximum-likelihood estimator for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013800/a0138009.png" />, which makes the classical definition meaningful. If the asymptotically-efficient estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013800/a01380010.png" /> exists, the magnitude
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A concept which extends the idea of an efficient estimator to the case of large samples (cf. [[Efficient estimator|Efficient estimator]]). An asymptotically-efficient estimator has not been uniquely defined. Thus, in its classical variant it concerns the asymptotic efficiency of an estimator in a suitably restricted class $\mathfrak K$ of estimators. In fact, let $T_n$ be a [[Consistent estimator|consistent estimator]] of a one-dimensional parameter $\theta$ constructed from a random sample of size $n$. Then $T_n\in\mathfrak K$ if the variance $\sigma^2(\sqrt nT_n)$ exists, and if it is bounded from below, as $n\to\infty$, by the inverse of the [[Fisher amount of information|Fisher amount of information]] corresponding to one observation. An estimator $T_n^*\in\mathfrak K$ which attains the lower bound just mentioned is asymptotically efficient. Under certain conditions this property is satisfied by the maximum-likelihood estimator for $\theta$, which makes the classical definition meaningful. If the asymptotically-efficient estimator $T_n^*$ exists, the magnitude
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013800/a01380011.png" /></td> </tr></table>
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$$\lim_{n\to\infty}\frac{\sigma^2(\sqrt nT_n^*)}{\sigma^2(\sqrt nT_n)}$$
  
is called the asymptotic relative efficiency of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013800/a01380012.png" />. Certain variants of the concept of an asymptotically-efficient estimator are due to R.A. Fisher, C.R. Rao and others.
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is called the asymptotic relative efficiency of $T_n$. Certain variants of the concept of an asymptotically-efficient estimator are due to R.A. Fisher, C.R. Rao and others.
  
 
====References====
 
====References====

Latest revision as of 10:57, 7 August 2014

A concept which extends the idea of an efficient estimator to the case of large samples (cf. Efficient estimator). An asymptotically-efficient estimator has not been uniquely defined. Thus, in its classical variant it concerns the asymptotic efficiency of an estimator in a suitably restricted class $\mathfrak K$ of estimators. In fact, let $T_n$ be a consistent estimator of a one-dimensional parameter $\theta$ constructed from a random sample of size $n$. Then $T_n\in\mathfrak K$ if the variance $\sigma^2(\sqrt nT_n)$ exists, and if it is bounded from below, as $n\to\infty$, by the inverse of the Fisher amount of information corresponding to one observation. An estimator $T_n^*\in\mathfrak K$ which attains the lower bound just mentioned is asymptotically efficient. Under certain conditions this property is satisfied by the maximum-likelihood estimator for $\theta$, which makes the classical definition meaningful. If the asymptotically-efficient estimator $T_n^*$ exists, the magnitude

$$\lim_{n\to\infty}\frac{\sigma^2(\sqrt nT_n^*)}{\sigma^2(\sqrt nT_n)}$$

is called the asymptotic relative efficiency of $T_n$. Certain variants of the concept of an asymptotically-efficient estimator are due to R.A. Fisher, C.R. Rao and others.

References

[1] C.R. Rao, "Linear statistical inference and its applications" , Wiley (1965)


Comments

More modern definitions of this concept are due to J. Hajek, L. LeCam and others.

References

[a1] J.A. Ibragimov, "Statistical estimation: asymptotic theory" , Springer (1981) (Translated from Russian)
How to Cite This Entry:
Asymptotically-efficient estimator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotically-efficient_estimator&oldid=32760
This article was adapted from an original article by O.V. Shalaevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article