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 225

observed data distribution. Locally ecient estimation of smooth functionals
of the distribution of (T;L) is presented in van der Laan and Robins (1998),
Andrews et al. (2005), and in Chapter 4 of van der Laan and Robins (2003).
The proposed methods are based on solving the optimal estimating equation
defined by the ecient influence curve of the statistical target parameter of
interest identifying this smooth parameter of the full data distribution. Es-
timation of the distribution function of T itself based on this data structure
is presented in van der Vaart and van der Laan (2006) and requires locally
ecient estimation of the primitive of the distribution function as well as the
iterative convex minorant algorithm. In the latter article, the asymptotic limit
distribution of the proposed estimator is also derived, directly generalizing the
limit distribution of the nonparametric maximum likelihood estimator for the
marginal current status data structure.
In the above described example of interval-censored data, a participant
may have died before ever being monitored. This type of application is also
covered by the extended current status data structure by defining L(t) to in-
clude the survival process I(T 0 t), truncating L(t) = L(min(t;T 0 )) at time
until death T 0 , and deterministically setting V equal to T 0 once death has
occurred; such an operation does not violate the coarsening at random as-
sumption. With this modification, the extended current status data structure
(V;I(T V ); L(V )) is equivalent to (min(V;T);I(V T);I(T 0 V ); L(V ));
it therefore also includes situations wherein certain survival times may have
been right-censored. The marginal version of this data structure has been stud-
ied in Dinse and Lagakos (1982), for example, and double robust estimating
equation methodology for the extended data structure is presented in Chapter
4 of van der Laan and Robins (2003). In longitudinal studies involving a hidden
time-to-event, a subject will commonly be repeatedly monitored. Groeneboom
and Wellner (1992) and Geskus and Groeneboom (1997) prove the eciency
of smooth functionals of the nonparametric maximum likelihood estimator for
the marginal interval-censored data structure with two monitoring times. An