Theory of Statistics - George Mason University

546 7 Statistical Hypotheses and Confidence Sets
S(x) ∋ θ ⇔ x ∈ A(θ),
can often be used to relate UMP invariant tests to best equivariant confidence
sets.
Equivariance for confidence sets is defined similarly to equivariance in
other settings.
Under the notation developed above, for the group of transformations G
and the induced transformation groups G ∗ and G, a confidence set S(x) is
equivariant if for all x ∈ X and g ∈ G,
g ∗ (S(x)) = S(g(x)).
The uniformly most powerful property of the test corresponds to uniformly
minimizing the probability that the confidence set contains incorrect values,
and the invariance corresponds to equivariance.
An equivariant set that is Θ-uniformly more accurate (“more” is defined
similarly to “most”) than any other equivariant set is said to be a uniformly
most accurate equivariant (UMAE) set.
There are situations in which there do not exist confidence sets that have
uniformly minimum probability of including incorrect values. In such cases,
we may retain the requirement for equivariance, but impose some other criterion,
such as expected smallest size (wrt Lebesgue measure) of the confidence
interval.
7.10 Asymptotic Confidence sets
It is often difficult to determine sets with a specified confidence coefficient or
significance level, or with other specified properties.
In such cases it may be useful to determine a set that “approximately”
meets the specified requirements.
What does “approximately” mean?
• uses numerical approximations
• uses approximate distributions
• uses a random procedure
• uses asymptotics
if
We assume a random sample X1, . . ., Xn from P ∈ P
An asymptotic significance level of a confidence set C(X) for g(θ) is 1 − α
lim inf Pr(C(X) ∋ θ) ≥ 1 − α for any P ∈ P.
n
The limiting confidence coefficient of a confidence set C(X) for θ is
lim inf Pr(C(X) ∋ θ)
n P ∈P
Theory of Statistics c○2000–2013 James E. Gentle