# Asymptotically-efficient estimator

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://www.encyclopediaofmath.org/index.php?title=Asymptotically-efficient_estimator&oldid=32760