Biomedical Engineering Reference
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jk
= Pr(Y = jjw = k). Then the logarithm transformation of the resulted
prole likelihood function has the form
l() = logL
1
() + logL
2
(;) = l
1
() + l
2
(;);
where
X
logL
1
() =
log f
(Y
i
jX
i
);
i2V
L
2
(;) =
2
X
X
log S
k
(X
i
)
2
X
X
2
n
jk
log
jk
2
X
X
logf1+
k
h
k
(X
i
)g
k=1
i2V
k
k=1
j=1
k=1
i2V
k
where
S
k
(X
i
) =
n
0k
n
k
+
n
1k
=n
k
1k
Pr(Y = 1jX
i
) +
n
2k
=n
k
1
1k
P(Y + 2jX
i
);
h
k
(X
i
) =
Pr(Y = 1jX
i
)
1k
S
k
(X
i
)
n
1k
=n
k
1k
+
n
2k
=n
k
1
1k
and
k
=
k
:
^
be the maximizer for l(). Denote
Let
Z
X
k
X
2
V (
0
) =
m
jk
(x;y;)
2
w
jk
(x;)dydG
k
(x)
k
jk
k=1
j=1
)
Z
2
m
jk
(x;y;)w
jk
(x;xi)dydG
k
(x)
U() =
1
n
Ef@
2
l()=@@
0
g;
where a
2
= aa
0
,
k
= n
k
=n and
jk
= n
jk
=n
k
as n!1, w
0k
(x;) = 1,
w
1k
(x;) = Pr(Y = 1jX)=
1k
and w
2k
= Pr(Y = 2jX)=(1
1k
),
0
@
@ log f
(yjx)
1
@S
k
(x)=@
S
k
(x)
k
@h
k
(x)=@
1+
k
h
k
(x)
@
A
@S
k
(x)=@
1k
S
k
(x)
k
@h
k
(x)=@
1k
1+
k
h
k
(x)
m
jk
(x;y;) =
d
jk
h
k
(x)
1+
k
h
k
(x)
with d
0k
= 0;d
1k
=1=
1k
and d
2k
= 1=(1
1k
).
The following theorem is due to Wang and Zhou
12
.
Theorem 3: Under general regularity conditions, n
1=2
(
^
0
)!
D
N(0; (
0
)) in a neighborhood of the true
0
= (;
11
;
12
;
21
;
22
; 0; 0)
0
,
where (
0
) = U
1
(
0
)V (
0
)U
1
(
0
). A consistent estimator of the
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