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382 7. Point Estimation

It has now become a common place for some authors of Bayesian papers
to claim that a proposed “Bayes estimator” or “Bayes decision” also satisfies
the “consistency” property in some sense. In the Bayesian literature, some
authors refer to this as the Bayesian-Frequentist compromise. Will it then be
fair to say that the (Fisherian) consistency property has not been such a bad
concept after all? The “history” does have an uncanny ability to correct its
own course from time to time! At the very least, it certainly feels that way.
A simple truth is that R. A. Fisher never suggested that one should choose
an estimator because of its consistency property alone. The fact is that in
many “standard” problems, some of the usual estimators such as the MLE,
UMVUE or estimators obtained via method of moments are frequently con-
sistent for the parameter(s) of interest.
The common sense dictates that the consistency property has to take a
back seat when considered in conjunction with the sufficiency property. The
estimator T defined in (7.7.1) is not sufficient for θ for any reasonable fixed-
n
sample-size n when k = 10 . On the other hand, the sample mean
6
is sufficient for θ and it also happens to be consistent for θ.
We should add that, in general, an MLE may not be consistent for ?.
Some examples of inconsistent MLE’s were constructed by Neyman and
Scott (1948) and Basu (1955b). In Chapter 12, under mild regularity condi-

tions, we quote a result in (12.2.3) showing that an MLE is indeed consis-
tent for the parameter θ it is supposed to estimate in the first place. This
result, in conjunction with the invariance property (Theorem 7.2.1), make the
MLE’s very appealing in practice.

7.8 Exercises and Complements

2
7.2.1 (Example 7.2.3 Continued) Suppose that X , ..., X are iid N(0, σ )
2
n
1
where q = σ is unknown, 0 < s < ¥. Here we have χ = Θ = ℜ and h ≡ h (q)
1
1
= E [X ] = 0, for all q. It is clear that the equations
q
1
given by the first and third moments in (7.2.1) do not lead to anything inter-
esting. In the Example 7.2.3 we had used the expression of η (θ). Now we
2
may arbitrarily move to the fourth moment and write
2
Using η , find an appropriate estimator of σ .
4
How is this estimator different from the one obtained in the Example 7.2.3?
7.2.2 (Exercise 6.3.5 Continued) Suppose that X , ..., X are iid dis-
1
n
tributed as Geometric(p) random variables with the common pmf given
x
by f(x; p) = p(1 - p) , x = 0, 1, 2, ... , and 0 < p < 1 is the unknown