Current Status Data in the Twenty-First Century: Some Interesting Developments - Interval-Censored-Time-to-Event Data: Methods and Applications - page 46

steadily in the recent past, and it would be dicult for me to do justice

to all the activity in this area in a chapter (of reasonable length) without
being cursory. I will concern myself primarily with some problems that are

closest to my own interests, describe some of the relevant results, and discuss

some open problems and conjectures. Before starting out, I would like to

acknowledge some of the topics and reviews in this area that I have found both

enlightening and useful: the topic on semiparametric information bounds and

nonparametric maximum likelihood estimation by Groeneboom and Wellner

(1992), the review by Huang and Wellner (1997), the review of current status

data by Jewell and van der Laan (2003), and last but not least, the topic on

Interval-censoring by Sun (2006).

The current status model is one of the most well-studied survival mod-

els in statistics. An individual at risk for an event of interest is monitored

at a particular observation time, and an indicator of whether the event has

occurred is recorded. An interesting feature of this kind of data is that the

NPMLE (nonparametric maximum likelihood estimator) of the distribution
function (F) of the event time converges to the truth at rate n 1=3 (n, as

usual, is the sample size) when the observation time is a continuous ran-

dom variable. Also, under mild conditions on the event-time distribution, the

(pointwise) limiting distribution of the estimator in this setting is the non-

Gaussian Cherno's distribution. This is in contrast to right-censored data

where the underlying survival function can be estimated nonparametrically
at rate p n under right-censoring and is pathwise norm-differentiable in the

sense of van der Vaart (1991), admitting regular estimators and normal lim-

its. On the other hand, when the status time in current status data has a

distribution with finite support, the model becomes parametric (multinomial)
and the event-time distribution can be estimated at rate p n. The current sta-

tus model, which goes back to Ayer et al. (1955), van Eeden (1956), and van

Eeden (1957), was subsequently studied by Turnbull (1976) in a more gen-

eral framework and asymptotic properties for the nonparametric maximum