Theory of Statistics - George Mason University

416 5 Unbiased Point Estimation
where, for each n, Vn(θ) is a k ×k positive definite matrix. From the definition
2 of asymptotic expectation of ˆθn − θ , Vn(θ) is the asymptotic variance-
covariance matrix of ˆ θn. Note that this matrix may depend on θ. We should
note that for any fixed n, Vn(θ) is not necessarily the variance-covariance
matrix of ˆ θn; that is, it is possible that Vn(θ) = V( ˆ θn).
Just as we have defined Fisher efficiency for an unbiased estimator of fixed
size, we define a sequence to be asymptotically Fisher efficient if the sequence
is asymptotically unbiased, the Fisher information matrix In(θ) exists and is
positive definite for each n, and Vn(θ) = (In(θ)) −1 for each n. The definition
of asymptotically (Fisher) efficiency is often limited even further so as to apply
only to estimators that are asymptotically normal. (MS2 uses the restricted
definition.)
Being asymptotically efficient does not mean for any fixed n that ˆ θn is
efficient. First of all, for fixed n, ˆ θn may not even be unbiased; even if it is
unbiased, however, it may not be efficient.
As we have emphasized many times, asymptotic properties are different
from limiting properties. As a striking example of this, consider a very simple
example from Romano and Siegel (1986).
Example 5.25 Asymptotic and Limiting Properties
Let X1, . . ., Xn iid ∼ N1(µ, 1), and consider a randomized estimator ˆµn of µ
defined by
⎧
⎨
ˆµn =
⎩
n2 with probability 1
n .
Xn with probability 1 − 1
n
It is clear that n 1/2 (ˆµn−µ) d → N(0, 1), and furthermore, the Fisher information
for µ is n −1/2 . The estimator ˆµn is therefore asymptotically Fisher efficient.
The bias of ˆµn, however, is
E(ˆµn − µ) = µ
which tends to infinity, and the variance is
V(ˆµn) = E(ˆµ 2 ) − (E(ˆµ)) 2
=
1 − 1
n
1 − 1
+ n − µ = n − µ/n,
n
1
n +
= n 3 + O(n 2 ),
1
n
n 4
− µ 1 − 1
2 + n
n
which also tends to infinity. Hence, we have an asymptotically Fisher efficient
estimator whose limiting bias and limiting variance are both infinite.
The example can be generalized to any estimator Tn of g(θ) such that
V(Tn) = 1/n and n 1/2 (Tn − g(θ)) d → N(0, 1). From Tn form the estimator
Theory of Statistics c○2000–2013 James E. Gentle