EXERCISES 21
(b) Show that, if H
0
is true, then pval
θ
0
(
¯
X), as a function of
¯
X, is Unif(0, 1).
(c) For given α ∈ (0,1), show that the test δ
0
, with
δ
0
(X
1
,.. . ,X
n
) =
(
1 if pval
θ
0
(
¯
X) ≤ α
0 if pval
θ
0
(
¯
X) > α,
is equivalent to the test δ derived in Exercise 1.8.
Exercise 1.10. A general method for constructing tests (and also confidence regions,
see Exercise 1.11) is based on likelihood ratios. Consider testing H
0
: θ = θ
0
versus
H
1
: θ 6= θ
0
, where θ
0
is some fixed value. Let L(θ ) be the likelihood function,
depending on the observed data X, and let
ˆ
θ be the maximum likelihood estimator.
Then the likelihood ratio statistic is
T = L(θ
0
)/L(
ˆ
θ).
Then the likelihood ratio test rejects H
0
iff T is less than a specified cutoff k.
(a) If the sampling model is X ∼ N(θ