Difference between revisions of "Asymptotically-efficient estimator"
(Importing text file) |
(TeX) |
||
| Line 1: | Line 1: | ||
| − | 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 | + | {{TEX|done}} |
| + | 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 | + | 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) |
Asymptotically-efficient estimator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotically-efficient_estimator&oldid=14672