Theory of Statistics - George Mason University

222 3 Basic Statistical Theory
A sufficient condition for the statistic T to be complete and sufficient for
P ∈ P is that q(P) contain an interior point.
Complete sufficiency is useful for establishing independence using Basu’s
theorem (see below), and in estimation problems in which we seek an unbiased
estimator that has minimum variance uniformly (UMVUE, discussed more
fully in Section 5.1).
It is often relatively straightforward to identify complete sufficient statistics
in certain families of distributions, such as those in the exponential class;
see Example 3.6. In a parametric-support family, there may be a complete
statistic. If so, it is usually an extreme order statistic; see Example 3.7.
Theorem 3.5 (minimal sufficiency III)
If T is a complete statistic in P and T is sufficient, then T is minimal sufficient.
Proof. Exercise (follows from definitions).
Complete sufficiency implies minimal sufficiency, but minimal sufficiency
does not imply completeness, as we see in the following example.
Example 3.4 minimal sufficient but not complete sufficient
Consider a sample X of size 1 from U(θ, θ+1). Clearly, X is minimal sufficient.
Any bounded periodic function h(x) with period 1 that is not a.e. 0 serves to
show that X is not complete. Let h(x) = sin(2πx). Then
E(h(X)) =
θ+1
θ
dx = 0.
Clearly, however h(X) is not 0 a.e., so X is not complete. We can see from
this that there can be no complete statistic in this case.
We will later define completeness of a class of statistics called decision
rules, and in that context, define minimal completeness of the class.
Basu’s Theorem
Complete sufficiency, ancillarity, and independence are related.
Theorem 3.6 (Basu’s theorem)
Let T(X) and U(X) be statistics from the population Pθ in the family P If
T(X) is a boundedly complete sufficient statistic for Pθ ∈ P, and if U(X) is
ancillary for Pθ ∈ P, then T and U are independent.
Proof.
If U is ancillary for Pθ and A is any set, then Pr(U ∈ A) is independent of
Pθ. Now, consider pA(t) = Pr(U ∈ A|T = t). We have
EPθ(pA(T)) = Pr(U ∈ A);
Theory of Statistics c○2000–2013 James E. Gentle