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Past exam of the mathematics course of the University of Cambridge
/
2021
/
iii
/
Paper 207
/
4
/
e
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Mathematics course of the University of Cambridge
Past exam of the mathematics course of the University of Cambridge
2021
iii
Paper 207
4
2026-09-28
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Solution
e
Solution
0
0
0
e
Put
D
j
=
H
(
x
j
)
−
H
(
x
j
−
1
)
, with
H
(
x
0
)
=
0
. Then
∑
i
=
1
n
H
(
x
i
)
=
∑
j
=
1
n
(
n
−
j
+
1
)
D
j
.
(1)
The natural local calibration is therefore
(
n
−
j
+
1
)
D
j
=
v
j
,
D
j
=
n
−
j
+
1
v
j
.
(2)
Taking
H
to be the right-continuous
step function
with these increments gives
H
(
t
)
=
∑
j
:
x
j
≤
t
n
−
j
+
1
v
j
.
(3)
Because
n
−
j
+
1
is exactly the
risk-set
size
r
j
, this is the estimator from part
c
and automatically satisfies
∑
i
{
v
i
−
H
(
x
i
)}
=
0
.
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4
Paper 207
iii
2021
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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