Current Status Data in the Twenty-First Century: Some Interesting Developments - Interval-Censored-Time-to-Event Data: Methods and Applications - page 77

3.6 Inference for Current Status Data on a Grid

While the literature on current status data is large, somewhat surprisingly, the

problem of making inference on the event time distribution, F, when the obser-

vation times lie on a grid with multiple subjects sharing the same observation

time had never been satisfactorily addressed. This important scenario, which

transpires when the inspection times for individuals at risk are evenly spaced

and multiple subjects can be inspected at any inspection time, is completely
precluded by the assumption of a continuous observation time. One can also

think of a situation where the observation times are all distinct but cluster into

a number of distinct well-separated clumps with very little variability among

the observation times in a single clump. For making inference on F in this

situation, the assumption of a continuous observation time distribution would

not be ideal, and a better approximation might be achieved by considering all

points within one clump to correspond to the same observation time|say, the

mean observation time for that clump. For simple current status data on a

regular grid, say with K grid-points, sample size n and n i individuals sharing
the i-th grid-point as the common observation time, how does one construct a

reliable confidence interval for the value of F at a grid-point of interest? What

F(t g ), where t g is

asymptotic approximations should the statistician use for

F the MLE? Some thought shows that this hinges critically

a grid-point and

on the size of n relative to K. If n is much larger than K and the number of

individuals per time is high, the problem can be viewed as a parametric one
and a normal approximation should be adequate. If n is \not too large" rela-

tive to K, the normal approximation would be suspect and the usual Chernoff

approximation may be more ideal. As Tang et al. (2012) show, one can view

this problem in an asymptotic framework and the nature of the approximation

depends heavily on how large K = K(n) is, relative to n. Unfortunately, the

rate of growth of K(n) is unknown in practice; Tang et al. (2012) suggest a