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6. Sufficiency, Completeness, and Ancillarity 320

Example 6.6.2 Suppose g(t; 0, σ) = exp {-½t /σ }, –∞ < t <
2
2
∞, σ ∈ ℜ . Is the family of distributions {g(t; 0, σ): σ > 0} complete? The
+
answer is “no,” it is not complete. In order to prove this claim, consider the
function h(t) = t and then observe that E [h(T)] = E [T] = 0 for all 0 < σ < ∞,
σ
σ
but h(t) is not identically zero for all t ∈ ℜ. Hence, the claim is true. !
6.6.1 Complete Sufficient Statistics

The completeness property of a statistic T looks like a mathematical con-
cept. From the statistical point of view, however, this concept can lead to
important results when a complete statistic T also happens to be sufficient.
Definition 6.6.4 A statistic T is called complete sufficient for an un-
known parameter θ if and only if (i) T is sufficient for θ and (ii) T is com-
plete.
In Section 6.6.3, we present a very useful theorem, known as Basu’s
Theorem, which needs a fusion of both concepts: sufficiency and com-
pleteness. Theorem 6.6.2 and (6.6.8) would help one to claim the minimality
of a complete sufficient statistic. Other important applications which ex-
ploit the existence of a complete sufficient statistic will be mentioned in later
chapters.

Example 6.6.3 Let X , ..., X be iid Bernoulli(p), 0 < p < 1. We already
1
n
know that is a sufficient statistic for p. Let us verify that T is
complete too. Since T is distributed as Binomial(n, p), the pmf induced by T is
given by g(t; p) = p (1 - p) , t ∈ T = {0, 1, ..., n}. Consider any real valued
t
n-t
function h(t) such that E [h(T)] = 0 for all 0 < p < 1. Now, let us focus on the
p
-1
possible values of h(t) for t = 0, 1, ..., n. With γ = p(1 - p) , we can write
From (6.6.4) we observe that E [h(T)] has been expressed as a polynomial of
p
the n degree in terms of the variable γ ∈ (0, ∞). A n degree polynomials in
th
th
? may be equal to zero for at most n values of γ ∈ (0, ∞). If we are forced to
assume that

then we conclude that that is for all t = 0,
1, ..., n. In other words, h(t) ≡ 0 for all t = 0, 1, ..., n, proving the complete-
ness of the sufficient statistic T. !