
66 INFERENTIAL MODELS
4.5.3.1 One-sided assertions
Here we consider a one-sided assertion, e.g., A = {θ ∈ Θ : θ < θ
0
}, where θ
0
is
fixed. This “left-sided” assertion is the kind we shall focus on, but other one-sided
assertions can be handled similarly. In this context, we can consider a very strong
definition of optimality.
Definition 4.1. Fix a left-sided assertion A. For two nested predictive random sets S
and S
0
, the IM based on S is said to be more efficient than that based on S
0
if, as func-
tions of X ∼P
X|θ
for any θ ∈A, R
A
(X; S) is stochastically larger than R
A
(X; S
0
). The
IM based on S
?
is optimal, or most efficient, if R
A
(X; S
?
) is stochastically lar ...