Current Status Data in the Twenty-First Century: Some Interesting Developments - Interval-Censored-Time-to-Event Data: Methods and Applications - page 69

Groeneboom et al. (2008b) compare the eciency of the MLE with respect
to the naive estimator and a scaled naive estimator, which makes a scal-
ing adjustment to the naive estimator when the sum of the components (i.e.
F 1 + F 2 ) exceeds one at some point (see Section 4 of their paper). It is seen
that the MLE is more ecient than its competitors, so the hard work in
computing and studying the MLE pays off. It should be noted that while
the MLE beats the naive estimators for the point-wise estimation of the sub-
distribution functions, estimates of smooth functionals of F 0i 's based on the
MLEs and the naive estimators are both asymptotically ecient|see, Jewell
et al. (2003) and Maathuis (2006). The discrepancy between the MLE and the
naive estimator therefore manifests itself only in the estimation of nonsmooth
functionals, like the value of the sub-distribution functions at a point.
Maathuis and Hudgens (2011) extend the work in the paper discussed
above to current status competing risks data with discrete or grouped obser-
vation times. In practice, recorded observation times are often discrete, mak-
ing the model with continuous observation times unsuitable. This led them
to investigate the limit behavior of the maximum likelihood estimator and
the naive estimator in a discrete model in which the observation time distri-
bution has discrete support, and a grouped model in which the observation
times are assumed to be rounded in the recording process, yielding grouped
observation times. They establish that the large sample behavior of the esti-
mators in the discrete and grouped models is critically different from that in
the smooth model (the model with continuous observation times); the max-
imum likelihood estimator and the naive estimator both converge locally at
p n rate and have limiting Gaussian distributions. The Gaussian limits in
their setting arise because they consider discrete distributions with a fixed
countable support and, in the case of grouping, a fixed countable number of
groups irrespective of sample size n. A similar phenomenon in the context
of simple current status data was observed by Yu et al. (1998). However, if
the support of the discrete distribution or the number of groupings (in the