Thus far we have discussed only the finite sample properties of estimators. It is frequently difficult, however, to obtain the exact moments, let alone the exact distribution, of estimators. In such cases we must obtain an approximation of the distribution or the moments. Asymptotic approximation is obtained by considering the limit of the sample size going to infinity. In Chapter 6 we studied the techniques necessary for this most useful approximation.
One of the most important asymptotic properties of an estimator is consistency.
DEFINITION 7.2.5 We say 0 is a consistent estimator of 0 if plim„^oo 0 = 0. (See Definition 6.1.2.)
In Examples 6.4.1 and 6.4.3, we gave conditions under which the sample mean and the sample variance are consistent estimators of their respective population counterparts. We can also show that under reasonable assumptions, all the sample moments are consistent estimators of their population values.
Another desirable property of an estimator is asymptotic normality. (See Section 6.2.) In Example 6.4.2 we gave conditions under which the sample mean is asymptotically normal. Under reasonable assumptions all the moments can be shown to be asymptotically normal. We may even say that all the consistent estimators we are likely to encounter in practice are asymptotically normal. Consistent and asymptotically normal estimators can be ranked by Definition 7.2.1, using the asymptotic variance in lieu of the exact mean squared error. This defines the term asymptotically better or asymptotically efficient.