Page 555 - Probability and Statistical Inference

532 11. Likehood Ratio and Other Tests

is assumed to have an exponential distribution with mean θ (> 0). The waiting
times for ten passengers were recorded as follows in one afternoon:

6.2 5.8 4.5 6.1 4.6 4.8 5.3 5.0 3.8 4.0
Test at 5% level whether the true average waiting time is any different from
5.3 minutes. {Hint: Use the test from Exercise 11.2.5.}
11.3.1 Verify the expressions of the MLE’s in (11.3.4).
11.3.2 Let the random variables X , ..., X be iid N(µ , ), i = 1, 2, and
ini
i
i1
that the X ’s be independent of the X ’s. Here we assume that µ , µ are
1j
2j
2
1
+
+
unknown and (µ , µ ) ∈ ℜ × ℜ but (σ , σ ) ∈ ℜ × ℜ are assumed known.
2
1
2
1
With preassigned α ∈ (0, 1), derive a level α LR test for H : µ = µ versus H 1
1
2
0
: µ ≠ µ in the implementable form. {Hint: The LR test rejects H if and only
1 2 0
if
11.3.3 Let the random variables X , ..., X be iid N(µ , σ ), i = 1, 2, and
2
i1
ini
i
that the X ’s be independent of the X ’s. Here we assume that µ , µ are
2
2j
1j
1
unknown and (µ , µ ) ∈ ℜ × ℜ but σ ∈ ℜ is assumed known. With preas-
+
1
2
signed α ∈ (0, 1), derive a level α LR test for H : µ = µ versus H : µ ≠ µ 2
1
1
1
2
0
in the implementable form. {Hint: The LR test rejects H if and only if
0
2
11.3.4 Let the random variables X , ..., X be iid N(µ, σ ), i = 1, 2, and that the
i
i1
ini
X ’s be independent of the X ’s. Here we assume that µ , µ , σ are all unknown and
1
2
1j
2j
(µ , µ ) ∈ ℜ × ℜ, σ ∈ ℜ . With preassigned α ∈ (0, 1) and a real number D, show
+
2
1
that a level α LR test for H : µ – µ = D versus H : µ – µ ≠ D would reject H
0 1 2 1 1 2 0
if and only if {Hint: Re-
peat the techniques from Section 11.3.1.}
11.3.5 (Exercise 11.3.2 Continued) Let the random variables X , ..., X be
i1 ini
iid N(µ , ), i = 1, 2, and that the X ’s be independent of the X ’s. Here we
2j
1j
i
assume that µ , µ are unknown and (µ , µ ) ∈ ℜ×ℜ but (σ , σ ) ∈ ℜ × ℜ +
+
2
2
1
1
1
2
are assumed known. With preassigned α ∈ (0, 1) and a real number D, derive
a level α LR test for H : µ – µ = D versus H : µ – µ ≠ D in the
2
1
2
1
0
1
implementable form. {Hint: The LR test rejects H if and only if
0
11.3.6 (Exercise 11.3.3 Continued) Let the random variables X , ...,
i1
X be iid N(µ , σ ), i = 1, 2, and that the X ’s be independent of the X ’s.
2
1j
ini
i
2j
Here we assume that µ , µ are unknown and (µ , µ ) ∈ ℜ × ℜ but σ ∈ ℜ +
2
1
2
1
is assumed known. With preassigned α ∈ (0, 1) and a real number D,
derive a level α LR test for H : µ – µ = D versus H : µ – µ ≠ D
0 1 2 1 1 2