Biomedical Engineering Reference
In-Depth Information
function of the observed ODS data is
Y
n
0
Y
K
Y
n
k
L(;G
X
) =
f
(Y
0i
;X
0i
)
f
(Y
ki
;X
ki
jY
ki
2C
k
):
i=1
k=1
i=1
By the Bayes' law, one can rewriteL(;G
X
) into
(
)
Y
n
0
Y
K
Y
n
k
jX
ki
)
Pr(Y
ki
2C
k
jX
ki
)
f
(Y
ki
L(;G
X
) =
f
(Y
0i
jX
0i
)
i=1
i=1
k=1
(
)
K
n
k
K
n
k
Y
Y
Y
Y
Pr(Y
ki
2C
k
jX
ki
)
Pr(Y
ki
g
X
(X
ki
)
2C
k
)
k=0
i=1
k=1
i=1
=L
1
()L
2
(;G
X
);
where
Z
Pr(Y
ki
2C
k
) =
Pr(Y
ki
2C
k
jx)g
X
(x)dx:
Obviously,L() is the conditional likelihood function based on the ob-
served ODS data.L(;G
X
) can be viewed as a marginal likelihood based
on (X
01
;:::;X
0n
0
;:::;X
11
;:::;X
1n
1
;:::;X
Kn
K
). For xed , this is an
extension of the biased sampling likelihood as discussed by Vardi
10;11
and
Qin
6
.
2.2. Algorithm and Asymptotics
The semiparametric empirical likelihood estimation for proposed by Zhou,
et al
15
can be obtained as follows.
First proleL
2
(;G
X
) by xing and obtaining an empirical likelihood
estimator G
X
(), over all discrete distributions whose support contains the
observed X values. This can be achieved by using the Lagrange multiplier
method.
The resulted prole likelihood function isL(; G
X
).
Use the Newton-Raphason procedure to maximize the resulted likelihood
fromL(; G
X
).
They illustrate the above algorithm with a simple setting corresponding
to a real study (the CPP in Zhou, et al.
15
).
K = 3, n
2
= 0;n
1
> 0;n
3
> 0 which corresponds to the Collaborative
Perinatal Project. Denote
1
= F(a
1
),
3
= 1F(a
2
), p
i
= g
X
(w
i
), where
(w
1
;:::;w
n
) = (X
01
;:::;X
0n
0
;X
11
;:::; X
1n
1
;X
31
;:::;X
3n
3
). Then, for a
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