# The Definition of Asymptotic Variance in Statistical Analysis

## An Introduction to Asymptotic Analysis of Estimators

The definition of the asymptotic variance of an estimator may vary from author to author or situation to situation. One standard definition is given in Greene, p 109, equation (4-39) and is described as "sufficient for nearly all applications." The definition for asymptotic variance given is:

asy var(t_hat) = (1/n) * limn->infinity E[ {t_hat - limn->infinity E[t_hat] }2 ]

## Introduction to Asymptotic Analysis

Asymptotic analysis is a method of describing limiting behavior and has applications across the sciences from applied mathematics to statistical mechanics to computer science. The term asymptotic itself refers to approaching a value or curve arbitrarily closely as some limit is taken. In applied mathematics and econometrics, asymptotic analysis is employed in the building of numerical mechanisms that will approximate equation solutions. It is a crucial tool in the exploration of the ordinary and partial differential equations that emerge when researchers attempt to model real-world phenomena through applied mathematics.

## Properties of Estimators

In statistics, an estimator is a rule for calculating an estimate of a value or quantity (also known as the estimand) based upon observed data. When studying the properties of estimators that have been obtained, statisticians make a distinction between two particular categories of properties:

1. The small or finite sample properties, which are considered valid no matter the sample size
2. Asymptotic properties, which are associated with infinitely larger samples when n tends to ∞ (infinity).

When dealing with finite sample properties, the aim is to study the behavior of the estimator assuming that there are many samples and as a result, many estimators. Under these circumstances, the average of the estimators should provide the necessary information. But when in practice when there is only one sample, asymptotic properties must be established. The aim is then to study the behavior of estimators as n, or the sample population size, increases. The asymptotic properties an estimator may possess include asymptotic unbiasedness, consistency, and asymptotic efficiency.

## Asymptotic Efficiency and Asymptotic Variance

Many statisticians consider the minimum requirement for determining a useful estimator is for the estimator to be consistent, but given that there are generally several consistent estimators of a parameter, one must give consideration to other properties as well. Asymptotic efficiency is another property worth consideration in the evaluation of estimators. The property of asymptotic efficiency targets the asymptotic variance of the estimators. Though there are many definitions, asymptotic variance can be defined as the variance, or how far the set of numbers is spread out, of the limit distribution of the estimator.