View source for 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 $\mathfrak K$ 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

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

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====
<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>