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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