
206 GENERALIZED INFERENTIAL MODELS
First, consider a hypothesis testing problem, H
0
: θ ∈ Θ
0
versus H
1
: θ 6∈ Θ
0
.
Define a plausibility function-based test as follows:
reject H
0
if and only if pl
y
(Θ
0
) ≤ α. (11.9)
The intuition is that if Θ
0
is not sufficiently plausible, given Y = y, then one should
conclude that the true θ is outside Θ
0
. An immediate consequence of Theorem 11.1
is that this test controls the probability of a Type I error at level α.
Corollary 11.1. For any α ∈ (0,1), the size of the test (11.9) is no more than α.
That is, sup
θ∈Θ
0
P
θ
{pl
Y
(Θ
0
) ≤ α} ≤ α. Moreover, if H
0
is a point-null, so that Θ
0
is a singleton, and T
Y,θ
is a continuous random variable ...