Biomedical Engineering Reference
In-Depth Information
l
2
(;) =
n
X
logf1 +
0
h(w
i
)g
n
X
logf(w
i
)gn
1
log
1
n
3
log
3
i=1
i=1
X
n
1
X
n
3
+
log F(a
1
jx
1j
) +
logf1F(a
2
jx
3k
)g
j=1
k=1
where h(w) = (h
1
(w);h
3
(w)), h
1
=fF(a
1
jw
i
)
1
g=(w
i
), h
3
=f1
g=(w
i
) and (w
i
) = k
0
+
k
1
1
F(a
1
jw
i
) +
k
3
3
F(a
2
jw
i
)
3
f1F(a
2
jw
i
)g:
Let
^
be the maximizer for l(). Dene
I
1
() 0
0
; I
1
() =
1
n
Ef@
2
l
1
=@@
0
g
V
2
() = k
0
Cov(e
0
) + k
1
Cov(e
1
) + k
3
Cov(e
3
); U() =
1
V
1
() =
0
n
Ef@
2
l()=@@
0
g;
where vectors e
0
;e
1
;e
3
are
0
1
0
1
@F (a
1
jx
1j
)
@
@(x
0j
)
@
1
(x
0i
)
@(x
0i
)
@
1
F (a
1
jx
1j
)
1
(x
1j
)
@
A
@
A
1
(x
0i
)
k
1
1
1
(x
1j
)
k
1
1
jx
1j
)
1
1
F(a
1
jx
0i
)
F(a
1
e
0i
=
; e
1j
=
F(a
2
jx
0i
)
h(x
0i
)
F(a
2
1
(x
0i
)
k
3
3
1
(x
1j
)
k
3
3
jx
1j
)
h(x
1j
)
and
0
@
1
A
@ F (a
2
jx
3k
)
@
@(x
3k
)
@
1
F (a
2
jx
3k
)
1
(x
3k
)
1
(x
3k
)
k
1
1
F(a
1
jx
3k
)
e
3k
=
:
F(a
2
jx
3k
)
1
3
h(x
3k
)
1
(x
3k
)
k
3
3
The following theorem is due to Zhou, et al.
15
.
Theorem 1: Under general regularity conditions, n
1=2
(
^
0
)!
D
N(0; (
0
)) in a neighborhood of the true
0
= (;
1
;
2
; 0; 0)
0
, where!
D
denotes convergence in distribution, (
0
) = U
1
(
0
)V (
0
)U
1
(
0
) and
V (
0
) = V
1
(
0
) + V
2
(
0
). A consistent estimator of the variance-covariance
matrix is U
1
(
^
) V (
^
) U
1
(
^
), where U and V are obtained by replacing the
large sample quantities in U and V with their corresponding small sample
quantities.
3. Semiparametric Estimated Likelihood for Two-Stage
ODS with a Continuous Outcome Variable
This section concerns statistical inference for ODS design where in addition
to the complete data considered in Section 2, some information about the
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