Biomedical Engineering Reference
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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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