Relationships among lim E, AE, and plim

Let F„ be the distribution function of X„ and Fn —■* Fat continuity points of F. We have defined plim Xn in Definition 3.2.1. We define lim E and AE as

(3.4.2)

In words, AE, which reads asymptotic expectation or asymptotic mean, is the mean of the limit distribution.

These three limit operators are similar but different; we can construct examples of sequences of random variables such that any two of the three concepts either differ from each other or coincide with each other. We shall state relationships among the operators in the form of examples and theorems. But, first, note the following obvious facts:

(i) Of the three concepts, only plim X„ can be a nondegenerate random variable; therefore, if it is, it must differ from lim EX„ or AE X„.

(ii) If plim Хп = а,& constant, then AE Xn = a. This follows immediately from Theorem 3.2.2.

Example 3.4.1. Let X„ be defined by

X„ = Z with probability (n — 1 )/n

= n with probability l/n,

where Z ~ N{0, 1). Then plim X„ = Z, lim EXn = 1, and AE Xn = EZ = 0.

Example 3.4.2. Let X„ be defined by

X„ = 0 with probability (n — 1 )/n

= n2 with probability l/n.

Then plim X„ = AE X„ = 0, and lim EX„ = lim n = °°.

Example 3.4.3. Let X~ N(a, 1) and Yn ~ N(P, и-1), where РФ 0. Then X/Yn is distributed as Cauchy and does not have a mean. Therefore lim E(X/Y„) cannot be defined either. But, because Yn /?, AE(X/Yn) = a/P

by Theorem 3.2.7 (iii).

The following theorem, proved in Rao (1973, p. 121), gives the conditions under which lim E = AE.

Theorem 3.4.1. If EXnr < M for all n, then lim EXsn = AE Xs„ for any s < r. In particular, if EX2 < M, then lim EXn ~ AEX„. (Note that this con­dition is violated by all three preceding examples.)

We are now in a position to define two important concepts regarding the asymptotic properties of estimators, namely, asymptotic unbiasedness and consistency.

Definition 3.4.1. The estimator 0„ of 0 is said to be asymptotically unbi­ased if AE 0„ = 0. We call AE 0„ — 0 the asymptotic bias.

Note that some authors define asymptotic unbiasedness using lim E instead of AE. Then it refers to a different concept.

Definition 3.4.2. The estimator 0„ of 0 is said to be a consistent estimator if plim 0„ = 0.

Some authors use the term weakly consistent in the preceding definition, to distinguish it from the term strong consistency used to describe the property

e^e.12

In view of the preceding discussions, it is clear that a consistent estimator is asymptotically unbiased, but not vice versa.