Biomedical Engineering Reference
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i.i.d. realizations from (;C;L), where = 1fT Cg. This is achieved via
estimating equations of the form
Z 1
(F;g;r)(c;;l) = r(c)(F(c j l) )
g(c j l)
+
r(s)F(s j l) ds;
0
for some real-valued function r dened on [0;1). Up to a constant, this is the
ecient influence function for estimating the functional R 1
0
r(s)S(s) ds in the
model where F(t j l) 1 F(t j l) and the distribution of L are left fully
unspecified. One estimate of S suggested by van der Vaart and van der Laan
is based on pure smoothing, namely,
S n;b (t) =P n (F n ;g n ;k b;t )
, where F n and g n are preliminary estimates of F and g, and k b;t (s) =
k((st)=b), where k is a probability density supported on [1; 1] and b b n is
a bandwidth that goes to 0 with increasing n. Th estimator S n;b (t) should be
viewed as estimating P F;g (F;g;k b;t ) = R 1
0 k b;t (s)S(s) ds, which converges
to S(t) as b ! 0. Under appropriate conditions on F n and g n as discussed
in Section 2.1 of that paper and which should not be dicult to satisfy, as
well as mild conditions on the underlying parameters of the model, Theorem
2.1 of van der Vaart and van der Laan (2006) shows that with b n = b 1 n 1=3 ,
n 1=3 (S n;b n (t) S(t)) converges to a mean 0 normal distribution. Sections 2.2
and 2.3 of the paper discuss variants based on the same estimating equa-
tion; while Section 2.2 relies only on isotonization, Section 2.3 proposes an
estimator combining isotonization and smoothing. This leads to estimators
with lower asymptotic variance than in Section 2.1, but there are caveats as
far as practical implementation is concerned and the authors note that more
refined asymptotics would be needed to understand the bias-variance trade-
off. Some discussion on constructing F n and g n is also provided, but there
is no associated computational work to illustrate how these suggestions work
for simulated and real data sets. It seems to me that there is scope here for
investigating the implementability of the proposed ideas in practice.
 
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