16 PRELIMINARIES
to use the “conditional distribution” of Z, given X = x, for the prediction problem.
However, given X = x, the distribution of Z is degenerate, i.e., Z = x −θ with prob-
ability 1. Since θ is unknown, this distribution is unknown, so Z is not conditionally
predictable. See the relevant section on fiducial inference in Chapter 2 for more dis-
cussion of this point. The remaining option is to use the “marginal distribution” of Z
for the prediction problem. That is, perform the prediction step based on the N(0, 1)
model, ignoring X entirely. This is the approach we advocate throughout the book.
Toward inference on θ, consider an interval assertion of the form
A = A
θ
0
,δ
= [θ
0
−δ ,θ
0
+ δ ],
where θ
0
∈ R and δ ≥0 are specified values. From the ...