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715
b.
∂
∂θ
log
L
(
θ

x
)
=
∂
∂θ
log
Y
i
log
θ
θ

1
θ
x
i
=
∂
∂θ
X
i
[loglog
θ

log(
θ

1) +
x
i
log
θ
]
=
X
i
±
1
θ
log
θ

1
θ

1
²
+
1
θ
X
i
x
i
=
n
θ
log
θ

n
θ

1
+
n
¯
x
θ
=
n
θ
³
¯
x

±
θ
θ

1

1
log
θ
²´
.
Thus,
¯
X
is the UMVUE of
θ
θ

1

1
log
θ
and attains the Cram´
erRao lower bound.
Note: We claim that if
∂
∂θ
log
L
(
θ

X
) =
a
(
θ
)[
W
(
X
)

τ
(
θ
)], then E
W
(
X
) =
τ
(
θ
), because
under the condition of the Cram´
erRao Theorem, E
∂
∂θ
log
L
(
θ

x
) = 0. To be rigorous, we
need to check the “interchange diﬀerentiation and integration“ condition. Both (a) and (b)
are exponential families, and this condition is satisﬁed for all exponential families.
7.39
E
θ
³
∂
2
∂θ
2
log
f
(
X

θ
)
´
=
E
θ
³
∂
∂θ
±
∂
∂θ
log
f
(
X

θ
)
²´
=
E
θ
"
∂
∂θ
µ
∂
∂θ
f
(
X

θ
)
f
(
X

θ
)
!#
= E
θ
∂
2
∂θ
2
f
(
X

θ
)
f
(
X

θ
)

µ
∂
∂θ
f
(
X

θ
)
f
(
X

θ
)
!
2
.
Now consider the ﬁrst term:
E
θ
"
∂
2
∂θ
2
f
(
X

θ
)
f
(
X

θ
)
#
=
Z ³
∂
2
∂θ
2
f
(
x

θ
)
´
d
x
=
d
dθ
Z
∂
∂θ
f
(
x

θ
)
d
x
(assumption)
=
d
dθ
E
θ
³
∂
∂θ
log
f
(
X

θ
)
´
= 0
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 Spring '12
 Dr.Hackney
 Statistics

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