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Stein's Paradox
Stein's Paradox
7 selected
Difficulty 3-8
7 unseen
View topic
Foundation
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1 foundation
4 intermediate
2 advanced
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Question 1 of 7
120s
foundation (3/10)
compute
Let
X
∼
N
(
θ
,
I
d
)
in
R
d
with
d
≥
3
. The MLE for
θ
is
θ
^
MLE
=
X
. What is its squared-error risk
E
∥
θ
^
MLE
−
θ
∥
2
?
Hide and think first
A.
The risk is
σ
2
d
/
n
where
n
is the sample size, by the standard variance formula for sample-mean Gaussian location estimators.
B.
The risk is at most
d
but generally smaller, since the MLE achieves the Cramér-Rao bound only at
θ
=
0
and improves elsewhere.
C.
Risk equals
d
for every
θ
since
X
−
θ
∼
N
(
0
,
I
d
)
has
E
∥
X
−
θ
∥
2
=
∑
i
Var
(
X
i
)
=
d
.
D.
Risk depends on
θ
: it is
∥
θ
∥
2
+
d
since bias contributes
∥
θ
∥
2
and variance contributes
d
to expected error.
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