Biomedical Engineering Reference
In-Depth Information
assumption stating, for each k 2 f1; 2;:::;m + 1g and a 2 f0; 1g, that there
exists some > 0 such that
pr(A(k) = a
0;a
(k)jL(k); A(k 1) = a
0;a
(k 1)) >
with probability 1 with a
0;a
(m + 1) dened, as before, as (a; 1; 1;:::; 1; (1; 1)),
and that pr(A(0) = ajL(0)) > with probability 1 for each a 2f0; 1g. Then,
the statistical parameter = (P) dened as (P) = Q
0
(P) Q
0
(P) and
estimable using the observed data is equivalent to the target parameter .
8.4.2
Estimation and Inference
We denote by o = (l(0);a(0);l(1);a(1);:::;a(m+ 1);l(m+ 2);y) the prototypi-
cal realization of a single observation. To build a targeted minimum loss-based
estimator for , and thus for under the causal assumptions listed above, we
Q
0
, Q
1
, ..., Q
a
m+2
.
first specify a sequence of appropriate loss functions for
Specically, we set, for k = 1; 2;:::;m + 2,
L
k;a
( Q
k
; Q
k+1
)(o) = I(a(k1) = a
0;a
(k1))
Q
k+1
(o) log Q
k
(o) + (1 Q
k+1
(o)) log(1 Q
k
(o))
;
where Q
a
m+3
(o) = y, and also set
Q
1
)(o) =
Q
1
(o) log
Q
0
(o) + (1 Q
1
(o)) log(1 Q
0
(o))
:
L
0;a
( Q
0
;
These loss functions satisfy, as required, that
Z
Q
k;0
= argmin
Q
k
L
k;a
( Q
k
;
Q
k+1;0
)(o)dP
0
(o)
Q
k;0
=
Q
k
(P
0
) is the true value of
Q
k
. The
for k = 0; 1;:::;m + 2, where
Q
0
can be written as D
a
=
P
m+2
ecient influence curve D
of
k=0
D
k;a
, where
for k 2f1; 2;:::;m + 2g
D
k;a
(O) =
I( A(k 1) = a
0;a
(k 1))
Q
k+1
Q
k
Q
k1
r=0
g
r
(a)
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