Theory of Statistics - George Mason University

538 7 Statistical Hypotheses and Confidence Sets
PrP (PS ∋ P) ≥ 1 − α ∀P ∈ P, (7.42)
for some given α ∈]0, 1[. The set PS is called a 1−α confidence set or confidence
set. The “confidence level” is 1 − α, so we sometimes call it a “level 1 − α
confidence set”. Notice that α is given a priori. We call
inf
P ∈P PrP (PS ∋ P) (7.43)
the confidence coefficient of PS.
If the confidence coefficient of PS is > 1 − α, then PS is said to be a
conservative 1 − α confidence set.
We generally wish to determine a region with a given confidence coefficient,
rather than with a given significance level.
If the distributions are characterized by a parameter θ in a given parameter
space Θ an equivalent 1 − α confidence set for θ is a random subset ΘS such
that
Prθ (ΘS ∋ θ) ≥ 1 − α ∀θ ∈ Θ. (7.44)
The basic paradigm of statistical confidence sets was described in Section
3.5.2, beginning on page 292. We first review some of those basic ideas,
starting first with simple interval confidence sets. Then in Section 7.9 we discuss
optimality of confidence sets.
As we have seen in other problems in statistical inference, it is often not
possible to develop a procedure that is uniformly optimal. As with the estimation
problem, we can impose restrictions, such as unbiasedness or equivariance.
We can define optimality in terms of a global averaging over the family
of distributions of interest. If the the global averaging is considered to be a
true probability distribution, then the resulting confidence intervals can be
interpreted differently, and it can be said that the probability that the distribution
of the observations is in some fixed family is some stated amount. The
HPD Bayesian credible regions discussed in Section 4.6.2 can also be thought
of as optimal sets that address similar applications in which confidence sets
are used.
Because determining an exact 1 − α confidence set requires that we know
the exact distribution of some statistic, we often have to form approximate
confidence sets. There are three common ways that we do this as discussed
in Section 3.1.4. In Section 7.10 we discuss asymptotic confidence sets, and in
Section 7.11, bootstrap confidence sets.
Our usual notion of a confidence interval relies on a frequency approach to
probability, and it leads to the definition of a 1 −α confidence interval for the
(scalar) parameter θ as the random interval [TL, TU], that has the property
Pr (TL ≤ θ ≤ TU) = 1 − α.
This is also called a (1 − α)100% confidence interval. The interval [TL, TU] is
not uniquely determined.
Theory of Statistics c○2000–2013 James E. Gentle