The second derivatives and their expected values for the simple Normal model
were derived in section 11.6 and yielded the following Fisher Information matrix
and the C-R lower bounds for any unbiased estimators of
and
2
:
I
(
θ
)=
μ
2
0
0
2
4
¶
(a)
C-R
(
)=
2
(b)
C-R
(
2
)=
2
4
In addition, the sampling distributions of the MLEs take the form (section 11.6):
(i)
b
v
N
(
2
)
(ii)
(
b
2
2
)
v
2
(
−
1)
(2.4.12)
b
is an
unbiased
,
fully e
ﬃ
cient
,
su
ﬃ
cient
, consistent,
asymptotically
Normal, asymptotically e
ﬃ
cient
estimator of
b
2
is
biased
,
su
ﬃ
cient
,
consistent
,
asymptotically Normal and asymptoti-
cally e
ﬃ
cient
.
Example 12.9
. Consider the simple
Gamma model
(table 12.10), with a density
function:
(
;
θ
)=
−
1
Γ
[
]
³
´
−
1
exp
n
−
³
´o
θ
:=(
)
∈
R
2
+
∈
R
+
22