Biomedical Engineering Reference
In-Depth Information
X. Let S denote the informative components of (W;Z), in the sense that
G(XjZ;W) = G(XjS) almost surely. Recognizing
X
K
X
L
G(xjs) =
kl
(s)G
kl
(xjs);
k=1
l=1
where
kl
(s) = Pr(Y = k;C = ljs) and G
kl
(xjs) = G(xjs;Y = k;C = l):
Further, a kernel estimator of
kl
(s) is given by
P
N
i=1
I(Y
i
= k;C
i
= l)K
h
(S
i
s)
^
kl
(s) =
P
N
i=1
K
h
(S
i
s)
and a kernel estimator of G
kl
(xjs) is given by
P
i2V
sl
I(X
i
x)K
h
(S
i
s)
G
kl
(xjs) =
P
i2V
sl
K
h
(S
i
s)
where K
h
() = K(=h)=h and h is the bandwidth. Then one can construct
a weighted kernel-based empirical distribution estimator for G(xjs),
X
L
X
K
G(xjs) =
^
lk
(s) G
lk
(xjs):
l=1
k=1
Accordingly, a weighted estimator for f
(Y
j
jZ
j
;W
j
) is
(
)
Z
X
K
X
L
f
(Y
j
jZ
j
;W
j
) =
^
kl
(S
j
) G
kl
(xjS
j
)
f
(Y
j
jx;Z
j
)d
:
k=1
l=1
Then the estimated log likelihood function has the form
X
K
X
L
X
X
K
X
L
X
^
L() =
log f
(Y
j
log f
(Y
i
jX
i
;Z
i
) +
jZ
j
;W
j
):
j2V
kl
k=1
l=1
i2V
kl
k=1
l=1
The proposed estimator
^
is the solution to the score equation
@
^
L()=@ = 0. The following theorem is due to Wang and Zhou
13
.
^
is asymptotically normal,
Theorem 4: Under some regularity conditions
p
N(
^
that is
)!
D
N(0; ) as N!1where
X
K
X
L
kl
= I
1
() +
kl
I
1
()
kl
()I
1
();
k=1
l=1
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