Biomedical Engineering Reference
In-Depth Information
where h(h
1
;h
2
;:::;h
q
)i represents the set of all functions that may be approxi-
mated arbitrarily well by linear combinations of components of (h
1
;h
2
;:::;h
q
),
then the targeted minimum loss-based estimator of
0
will be locally ecient
and will generally enjoy certain robustness properties relative to model mis-
specification.
A more general, component-wise targeted minimum loss-based estimation
procedure can often be devised when the parameter Q of P upon which
depends can be written as a function of other parameters (Q
1
;Q
2
;:::;Q
J
),
so that (Q) =
1
(Q
1
;Q
2
;:::;Q
J
) for some parameter
1
. Suppose that, for
each j 2 f1; 2;:::;Jg, there exists a loss function L
j
: (
G
j
) R
m
!R
+
Q
j
;
such that
Z
Q
j;0
= argmin
Q
j
2
Q
j
L
j
(Q
j
;g
j;0
)(o)dP
0
(o)
where
G
j
= fg
j
(P) : P 2Mg, Q
j;0
= Q
j
(P
0
), g
j
is
a nuisance parameter, and g
j;0
= g
j
(P
0
). Suppose further that, for each j 2
f1; 2;:::;Jg and given Q
j
2
Q
j
= fQ
j
(P) : P 2Mg,
Q
j
with
finite-dimensional parameter
j
and typical element Q
j
(
j
) can be constructed
such that Q
j
(0) = Q
j
. If estimators Q
j;n
and g
j;n
of Q
j;0
and g
j;0
are available
for each j 2f1; 2;:::;Jg, then a variety of estimate uctuation schemes can be
Q
j
, a fluctuation sub-model Q
j
(Q
j
)
constructed whereby, in a particular ordering, the current estimate of each Q
j
is updated using the corresponding fluctuation sub-model and loss function.
In each of these steps, estimates of the nuisance parameter g
j
, that will often
depend on some components of (Q
1
;Q
2
;:::;Q
J
), may also be updated based
on current estimates of the latter. The targeted minimum loss-based estimator
presented below is an example of such. There will generally exist many possible
iterative updating schemes that will define different targeted minimum loss-
based estimators; under certain regularity conditions, however, these different
estimators will usually exhibit identical asymptotic behavior. In particular, if
the ecient influence curve D
(Q;g) of (Q) can be shown to be an element
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