Page 342 - Probability and Statistical Inference

6. Sufficiency, Completeness, and Ancillarity 319

holds. Consider any real valued function h(t) defined for t ∈ T, having finite
expectation, such that

In other words,

Definition 6.6.3 A statistic T is said to be complete if and only if the
family of distributions {g(t; θ): θ ∈ Θ} induced by T is complete.
In these definitions, observe that neither the statistic T nor the parameter θ
has to be necessarily real valued. If we have, for example, a vector valued
statistic T = (T , T , ..., T ) and θ is also vector valued, then the requirement
2
1
k
in (6.6.1) would obviously be replaced by the following:

Here, h(t) is a real valued function of t = (t , ..., t ) ∈ T and g(t; θ) is the pmf
1
k
or the pdf of the statistic T at the point t.
The concept of completeness was introduced by Lehmann and Scheffé
(1950) and further explored measure-theoretically by Bahadur (1957). Next
we give two examples. More examples will be given in the subsection
6.6.1.
Example 6.6.1 Suppose that a statistic T is distributed as Bernoulli(p), 0
< p < 1. Let us examine if T is a complete statistic according to the Defini-
tion 6.6.3. The pmf induced by T is given by g(t; p) = p (1 - p) , t = 0, 1.
1-t
t
Consider any real valued function h(t) such that E [h(T)] = 0 for all 0 < p <
p
1. Now, let us focus on the possible values of h(t) when t = 0, 1, and write

Observe that the middle step in (6.6.3) is linear in p and so it may be zero
for exactly one value of p between zero and unity. But we are demanding
that p{h(1) - h(0)} + h(0) must be zero for infinitely many values of p in
(0, 1). Hence, this expression must be identically equal to zero which
means that the constant term as well as the coefficient of p must indi-
vidually be both zero. That is, we must have h(0) = 0 and h(1) - h(0) =
0 so that h(1) = 0. In other words, we have h(t) = 0 for t = 0, 1. Thus,
T is a complete statistic. !