Theory of Statistics - George Mason University

7.8 Confidence Sets 541
For any given θ0 ∈ Θ, consider the nonrandomized test Tθ0 for testing the
simple hypothesis H0 : θ = θ0, against some alternative H1. We let A(θ0) be
the set of all x such that the test statistic is not in the critical region; that is,
A(θ0) is the acceptance region.
Now, for any θ and any value x in the range of X, we let
C(x) = {θ : x ∈ A(θ)}.
For testing H0 : θ = θ0 at the α significance level, we have
that is,
sup Pr(X /∈ A(θ0) | θ = θ0) ≤ α;
1 − α ≤ inf Pr(X ∈ A(θ0) | θ = θ0) = inf Pr(C(X) ∋ θ0 | θ = θ0).
This holds for any θ0, so
inf
P ∈P PrP(C(X) ∋ θ) = inf
θ0∈Θ inf PrP(C(X) ∋ θ0 | θ = θ0)
≥ 1 − α.
Hence, C(X) is a 1 − α level confidence set for θ.
If the size of the test is α, the inequalities are equalities, and so the confidence
coefficient is 1 − α.
For example, suppose Y1, Y2, . . ., Yn is a random sample from a N(µ, σ2 )
distribution, and Y is the sample mean.
To test H0 : µ = µ0, against the universal alternative, we form the test
statistic
n(n − 1)(Y − µ0)
T(X) =
Yi
− Y 2 which, under the null hypothesis, has a Student’s t distribution with n − 1
degrees of freedom.
An acceptance region at the α level is
t(α/2), t(1−α/2) ,
and hence, putting these limits on T(X) and inverting, we get
Y − t(1−α/2) s/ √ n, Y − t(α/2) s/ √
n ,
which is a 1 − α level confidence interval.
The test has size α and so the confidence coefficient is 1 − α.
Randomized Confidence Sets
To form a 1 − α confidence level set, we form a nonrandomized confidence
set (which may be null) with 1 − α1 confidence level, with 0 ≤ α1 ≤ α, and
then we define a random experiment with some event that has a probability
of α − α1.
Theory of Statistics c○2000–2013 James E. Gentle