Biomedical Engineering Reference
In-Depth Information
we denote this relevant part of P by Q(P). The parameter (P) is sometimes
written as (Q) to emphasize the dependence of on Q alone. Dene
=
fQ(P) : P 2 Mg to be the class of all possible Q under M. Second, a loss
Q
) R
m
!R
+
function L : (
Q
;
G
such that
Z
Q
0
= argmin
Q2
Q
L(Q; g
0
)(o)dP
0
(o)
should be constructed, where Q
0
= Q(P
0
) and g
0
= g(P
0
), with g(P) defined
to be some nuisance parameter taking values in
G
= fg(P) : P 2Mg. Finally,
for each given Q 2
Q, a uctuation sub-model, say Q(Q) = fQ() 2
Q: kk <
g for some > 0 and a nite-dimensional parameter , satisfying Q(0) = Q,
is required. Suppose estimators Q
n
of Q
0
and g
n
of g
0
are available. Then,
the TMLE algorithm is defined recursively as follows: given an estimator Q
n
of Q
0
, set
X
Q
k+1
n
= argmin
Q2Q(Q
n
)
L(Q; g
n
)(O
i
) ;
i=1
repeat until convergence, and take
n
= (Q
n
), where Q
n
= lim
k!1
Q
n
, to
be the targeted minimum loss-based estimator of
0
.
If the initial estimator Q
n
is consistent for Q
0
, then the consistency of
n
is guaranteed. However, even if Q
n
fails to be consistent for Q
0
, it may
be possible, in many instances, that
n
is consistent for
0
provided fluctu-
ation sub-models are cleverly constructed. The targeted minimum loss-based
estimator Q
n
of Q
0
will satisfy the estimating equation
Z
d
d
L(Q
n
(); g
n
)(o)
=0
dP
n
(o) = 0
where P
n
is the empirical distribution based on observations O
1
;O
2
;:::;O
n
.
This equation is the basis for the study of the weak convergence of
p
n(
n
0
).
If the fluctuation model is constructed in such a manner as to ensure that the
ecient influence curve D
(Q;g) of (Q) is contained in the closure of the
linear span of the loss-based score at = 0, that is,
=0
d
d
L(Q(); g)
D
(Q;g) 2
;
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