Biomedical Engineering Reference
In-Depth Information
where is the information matrix of .
^
n
is asymptotically ecient be-
cause its variance-variance matrix asymptotically achieves the information
lower bound. To get the estimation of the variance-covariance matrix, we de-
ne
S
0
(cjz)
1 S
0
(cjz)
O(cjz) =
and
exp[
0
(c)e
z
0
]
1 exp[
0
(c)e
z
0
]
;
R(c;z) =
2
(cjz)O(cjz) = e
2z
0
0
0
(c)
then the information for is
(
2
)
Z
E(ZR(C;Z)jC)
E(R(C;Z)jC)
= E
R(C;Z)
;
0
=
^
n
;
0
=
^
n
; then for con-
tinuous Z, E(R(C;Z)jC) can be approximated by E( R
n
(c;Z)jC = c), which
can be estimated by nonparametric methods. However, due to the complica-
where a
2
= aa
0
for a 2 R
p
. Let R
n
(c;z) = R(c;z)
tion of E(R(C;Z)jC) and E(ZR(C;Z)jC), there is no general approach for
the estimation of the information matrix, especially when Z is categorical.
For the simple case where Z is dichotomous, Huang (1996) showed that the
estimator of the information matrix of
0
has the form
n
R
n
(C
i
;Z
i
)[Z
i
^
n
(C
i
)]
2
o
X
1
n
^
n
=
i=1
with
R
n
(c; Z = 1) f
1
(c)n
1
R(c; Z = 1) f
1
(c)n
1
+ R
n
(c; Z = 0) f
0
(c)n
0
^
n
(c) =
:
Here, n
1
and n
0
are the number of subjects with Zi
i
= 1 and Z
i
= 0, respec-
tively; f
1
f1(c) and f
0
f1(c) are kernel estimators of the density functions f
1
(c) and
f
0
f0(c) for Ci
i
with Zi
i
= 1 and Z
i
= 0, respectively.
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