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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