Biomedical Engineering Reference
In-Depth Information
of
*
@
@
1
L
1
(Q
1
(
1
);g
1
)
!+
1
=0
J
=0
@
@
J
L
J
(Q
J
(
J
);g
J
)
;:::;
then the resulting targeted minimum loss-based estimator
n
of
0
will be
asymptotically ecient and will generally enjoy a certain level of robustness
to model misspecification.
8.3.2
Implementation of TMLE
8.3.2.1
Basic Ingredients
As suggested in van der Laan and Gruber (2011), we outline here a targeted
minimum loss-based estimator defined sequentially. This estimator will be
constructed by combining targeted minimum loss-based estimators developed
for each summand in the denition of (P). Denote a typical data realization
by o = (l(0);a(0);l(1);a(1);l(2);y). Specically, we consider the loss functions
L
2;a
( Q
2
)(o)
= I(a(1) = 1;a(0) = a)
y log
Q
2
(o) + (1 y) log(1 Q
2
(o))
;
L
1;a
( Q
1
;
Q
2
)(o)
= I(a(0) = a)
Q
2
(o) log
Q
1
(o) + (1 Q
2
(o)) log(1 Q
1
(o))
;
=
Q
1
(o) log
Q
0
(o) + (1 Q
1
(o)) log(1 Q
0
(o))
;
L
0;a
( Q
0
;
Q
1
)(o)
which can be verified to be such that
Z
Q
2;0
L
2;a
( Q
2
)(o)dP
0
(o) ;
=
argmin
Q
a
2
Z
Q
1;0
L
1;a
( Q
1
; Q
2;0
)(o)dP
0
(o) ;
=
argmin
Q
a
1
Z
Q
0;0
L
0;a
( Q
0
; Q
1;0
)(o)dP
0
(o) ;
=
argmin
Q
a
0
where Q
2;0
= Q
2
(P
0
), Q
1;0
= Q
1
(P
0
), and Q
0;0
= Q
0
(P
0
). The ecient
influence curve D
(Q;g) of Q
0
can be expressed as D
a
= D
2;a
+ D
1;a
+ D
0;a
,
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