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Past exam of the mathematics course of the University of Oxford
/
2023
/
Prelims
/
Mathematics III
/
7
/
b
/
i
Show body
Body
0
Past exam of the mathematics course of the University of Oxford
/
2023
/
Prelims
/
Mathematics III
/
7
/
b
/
ii
/
Solution
2026-10-25
View more
The inverse
moments
in part
i
give
E
[
θ
^
]
=
n
−
1
n
θ
,
E
[
θ
^
2
]
=
(
n
−
1
)
(
n
−
2
)
n
2
θ
2
.
(1)
Thus the
bias
is positive, and direct expansion of the
mean squared error
yields
b
θ
(
θ
^
)
=
n
−
1
θ
,
MSE
θ
(
θ
^
)
=
(
n
−
1
)
(
n
−
2
)
(
n
+
2
)
θ
2
.
(2)
For comparison, the
variance
is
n
2
θ
2
/
[(
n
−
1
)
2
(
n
−
2
)]
; adding the squared
bias
gives the same answer by the
bias-variance decomposition
.
Total
articles
:
1