Targeted Minimum Loss-Based Estimation of a Causal E ect Using Interval-Censored Time-to-Event Data - Interval-Censored-Time-to-Event Data: Methods and Applications - page 212

fore, that the targeted minimum loss-based estimator of 0 is

h Q 1;

1;n (O i ) i :

X

1
n

Q 1;

0;n Q 0; 1;n (O i ) Q 0;

n =

=

0;n

i=1

In this case, additional targeting steps do not result in further fluctuation from
the above estimates, and the algorithm is complete after a single step. In other

words, one round of targeting suces to achieve maximal bias reduction.

8.3.3 Asymptotic Results

The methodology proposed above is an example of a double-robust estimation
procedure. Specifically, the estimator n constructed will be consistent for the
statistical parameter of interest, provided either the initial estimators Q 2;n ,
Q 1;n , and Q 0;n are consistent for Q 2;0 , Q 1;0 , and Q 0;0 , respectively, or the
estimators g 0n (a) and g 1n (a) are consistent for g 0 (a) and g 1 (a), respectively.
A certain level of misspecification of working models used to estimate the

various ingredients required in the procedure is allowed without sacrificing

the consistency of the overall procedure; in practice, this is a particularly

useful property.
The estimator n constructed using the methodology presented is asymp-
totically linear; this follows directly from the fact that (i) n = (Q n ) with
a pathwise differentiable parameter, (ii) Q n solves the ecient influence curve
estimating equation

X

1
n

D (Q n ;g n )(O i ) = 0

i=1

where D = D 1 D 0 , Q n is the targeted minimum loss-based estimator of
Q, and g n = (g 0n ;g 1n ), and from (iii) required regularity conditions explicitly
stated in van der Laan and Rose (2011), including, in particular, the consis-
tency of at least one of g n and Q n . In particular, when g is known exactly so
that g n = g 0 , the influence curve of n is given by

IC 0 = D (Q 0 ;g 0 )