Page 404 - Probability and Statistical Inference

7. Point Estimation 381

it will help to think of V as a sequence of degenerate random variables. Then,
n
one can repeatedly apply the Theorems 5.2.4-5.2.5 and conclude that
and hence as n → ∞. That is,
is a consistent estimator of e .
-θ
One should also verify that the estimators W, W′, and T*(α*) defined via
(7.6.4)-(7.6.5) are also consistent for the unknown parameter θ in the
2
N(θ, θ ) population. Details are left out as Exercise 7.7.5.
Having found a number of estimators in so many statistical models which
are consistent for the parameters of interest, we wish to emphasize again that
consistency is after all a large sample property. In statistics, one invariably
works with data of fixed size n and hence one should not rely upon the con-
sistency property of an estimator alone to claim any superior performance on
behalf of the estimator under consideration.
D. Basu, among others, emphasized the limited usefulness of the concept
of consistency of an estimator. Basu gave the following forceful example. Let
X , ..., X be iid N(θ, 1) where θ ∈ (−∞, ∞) is the unknown parameter. Define
n
1
a sequence of estimators of θ as follows:

where k is fixed but presumably very large. Since is consistent
for θ, it follows from the definition of the estimator T that it would be consis-
n
tent for θ too. But, Basu would argue that in real life how many times does
one encounter a sample of size n larger than a million! So, he would focus on
the estimator T when k = 10 . Now, if an experimenter is committed to use
6
n
the estimator T from (7.7.1), then for all practical purposes, regardless of
n
what the observed data dictates, one will end up guessing that θ is zero and
yet such an estimator would be consistent for θ ! This construction of T n
unfortunately created an impression that there was something inherently wrong
with the concept of consistency. Can a reader think of any practical scenario
where T defined by (7.7.1) will be used to estimate θ?
n
Basu wanted to emphasize that the consistency of an estimator is not a
useful property in the practice of statistics. But, on the other hand, the given
T will almost certainly never be used in practice, and so it remains vague
n
what this example actually conveys. What this example might have demon-
strated was this: The consistency of an estimator may not be a useful prop-
erty in itself in the practice of statistics. This is exactly what we had said
before we gave Basu’s example! In other words, Basu’s example has not been
nearly as damaging to the concept of consistency as some non-Fisherians
may have liked to believe.