Statistics Graduate Level Quiz
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PROOF, DERIVATION & PROBLEM-SOLVING TOOLS
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PASSAGE
Suppose
n
(
θ
^
n
−
0
)
⇒
N
(
0
,
σ
2
)
\sqrt n(\widehat\theta_n-0)\Rightarrow N(0,\sigma^2)
n
(
θ
n
−
0
)
⇒
N
(
0
,
σ
2
)
with
σ
2
>
0
\sigma^2>0
σ
2
>
0
. The parameter of interest is
g
(
θ
)
=
θ
2
g(\theta)=\theta^2
g
(
θ
)
=
θ
2
.
Which conclusion correctly accounts for the fact that
g
′
(
0
)
=
0
g'(0)=0
g
′
(
0
)
=
0
?
A
n
g
(
θ
^
n
)
⇒
N
(
0
,
4
σ
2
)
\sqrt n\,g(\widehat\theta_n)\Rightarrow N(0,4\sigma^2)
n
g
(
θ
n
)
⇒
N
(
0
,
4
σ
2
)
by the ordinary delta method.
B
n
g
(
θ
^
n
)
⇒
N
(
0
,
σ
4
)
n\,g(\widehat\theta_n)\Rightarrow N(0,\sigma^4)
n
g
(
θ
n
)
⇒
N
(
0
,
σ
4
)
because the convergence rate is squared.
C
n
g
(
θ
^
n
)
⇒
σ
2
χ
1
2
n\,g(\widehat\theta_n)\Rightarrow\sigma^2\chi_1^2
n
g
(
θ
n
)
⇒
σ
2
χ
1
2
by applying the continuous mapping theorem to the squared limit.
D
No rescaling can produce a nondegenerate limit because the first derivative of
g
g
g
vanishes.
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