Biomedical Engineering Reference
In-Depth Information
g
r
(a) denotes the probability pr(A(r) = a
0;a
(r)jL(r); A(r 1) = a
0;a
(r 1))
for r 2f0; 1;:::;m + 1g, and D
0;a
(O) = Q
1
(Q). Using knowledge of these
ecient influence curves, we may construct optimal fluctuation sub-models
Q
0
,
Q
1
, ...,
Q
a
m+2
: specifically, we set
for components
!
logit Q
k
+
I( A(k 1) = a
0;a
(k 1))
Q
k
() = expit
Q
k1
r=0
g
r
(a)
for k 2 f1; 2;:::;m + 2g, Q
0
() = expit
logit Q
0
+
, and consider the sub-
models Q
k
( Q
k
) = f Q
k
() : jj < 1g. These sub-models are constructed to
ensure that, indeed,
=0
d
d
L
k;a
( Q
k
();
Q
k+1
)
= D
k;a
d
L
m+2;a
( Q
a
m+2
())
=0
= D
m+2;a
.
Suppose that for k 2 f0; 1;:::;m + 2g, initial estimators Q
k;n
of Q
k
and
g
k;n
of g
k
are available. Considered fluctuations of these estimators are of the
form
d
for k 2f0; 1;:::;m + 1g, and
!
logit Q
k;n
+
I( A(k 1) = a
0;a
(k 1))
Q
k;n
() = expit
Q
k1
r=0
g
r;n
(a)
for k 2 f1; 2;:::;m + 2g and Q
0;n
() = expit
logit Q
0;n
+
. The one-step
targeted estimate of Q
k
is defined as Q
a;
k;n
= Q
k;n
(
a;
k;n
) for k 2f0; 1;:::;m+2g,
where
a;
m+2;n
= argmin
L
m+2;a
( Q
a
m+2;n
()) and
L
k;a
( Q
k;n
(); Q
a;
a;
k;n
= argmin
k+1;n
)(O
i
)
for k 2f0;:::;m + 1g, resulting in the targeted estimate
h
Q
1;
i
X
1
n
Q
1;
0;n
Q
0;
1;n
(O
i
) Q
0;
n
=
0;n
=
1;n
(O
i
)
i=1
of
0
and, under the causal assumptions listed above, of
0
as well. As before,
the minimizer
a;
k;n
may be obtained directly by resorting to standard statistical
software. Specifically, we can obtain
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