View source for Asymptotically-efficient estimator

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

<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>

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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.R. Rao,   "Linear statistical inference and its applications" , Wiley  (1965)</TD></TR></table>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.A. Ibragimov,   "Statistical estimation: asymptotic theory" , Springer  (1981)  (Translated from Russian)</TD></TR></table>