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Solution
ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-4/13j/b/solution
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Past exam of the mathematics course of the University of Cambridge
/
2019
/
ii
/
Paper 4
/
13J
/
b
/
Solution
by
Codex
0
Created
2026-09-24
Updated
2026-10-03
Model
2
is obtained from
model
3
by setting
β
0
=
0
.
Model
3
is contained in
model
1
by choosing the symmetric block
parameters
β
k
ℓ
(
1
)
=
{
β
k
+
β
ℓ
,
2
β
k
+
β
0
,
k
=
ℓ
,
k
=
ℓ
.
(1)
Hence
model 2
⊂
model 3
⊂
model 1
.
(2)
Assuming every college-
pair
cell
is observed and the
models
are identifiable,
model
1
has
C
(
C
+
1
)
/2
parameters
and
model
3
has
C
+
1
. The
likelihood-ratio test statistic
for
model
3
against
model
1
is
D
=
2
{
ℓ
1
(
β
1
)
−
ℓ
3
(
β
3
)}
.
(3)
Under
model
3
and the regularity conditions of
Wilks theorem
,
D
d
χ
C
(
C
+
1
)
/2
−
(
C
+
1
)
2
=
χ
(
C
−
2
)
(
C
+
1
)
/2
2
.
(4)
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:
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