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8. Tests of Hypotheses 399

Let us, however, discuss few other notions for the moment. Sometimes
instead of focussing on the Type II error probability β, one considers what is
known as the power of a test.
Definition 8.2.1 The power or the power function of a test, denoted by
Q(θ), is the probability of rejecting the null hypothesis H when θ ∈ Θ is the
0
true parameter value. The power function is then given by

In a simple null versus simple alternative testing situation such as the one
we had been discussing earlier, one obviously has Q(θ ) = α and Q(θ ) =
0
1
1 - β.
Example 8.2.3 (Example 8.2.1 Continued) One can verify that for the
Test #4, the power function is given by Q(θ) = 1 - Φ (22.5 - 3θ) for all θ
∈ ℜ.
A test is frequently laid out as in (8.2.1) or (8.2.2). There is yet another
way to identify a test by what is called a critical function or test function.
Recall that proposing a test is equivalent to partitioning the sample space χ n
into two parts, critical region R and its complement R .
c
Definition 8.2.2 A function Ψ (.) : χ → [0, 1] is called a critical function
n
or test function where ψ(X) stands for the probability with which the null
hypothesis H is rejected when the data X = x has been observed,
0
n
x ∈ χ .
In general, we can rewrite the power function defined in (8.2.4) as follows:

8.2.1 The Concept of a Best Test

In (8.2.3) we pointed out how Type I and II error probabilities are evaluated
when testing between a simple null and simple alternative hypotheses. When
the null and alternative hypotheses are composite, we refocus on the defini-
tion of the two associated error probabilities. Let us consider testing H : θ ∈
0
Θ versus H : θ ∈ Θ where Θ ⊂ Θ, Θ ⊂ Θ and Θ ∩ Θ = ϕ, the empty
0
0
1
0
1
1
1
set.
Definition 8.2.3 We start with a fixed number α ∈ (0, 1). A test for H : θ
0
∈ Θ versus H : θ ∈ Θ with its power function Q(θ) defined in (8.2.2) is
1
1
0
called size a or level a according as Q(θ) = α or ≤ α respec-
tively.