Current Status Data in the Twenty-First Century: Some Interesting Developments - Interval-Censored-Time-to-Event Data: Methods and Applications - page 78

way to circumvent this problem by using a family of \boundary distributions"
indexed by a scale parameter c > 0, which provides correct approximations to
the centered and scaled MLE: n 1=3 ( F(t g ) F(t g )). Below, I briefly describe
the proposed method. Let [a;b] (a 0) be the interval on which the time
grid fa + ;a + 2 ;:::;a + Kg is dened (K is the smallest integer such that
a+(K+1) > b) and let n i be the number of individuals whose inspection time
is a+i. The MLE, F, is easily obtained as the solution to a weighted isotonic
regression problem with the n i 's acting as weights. Now, nd c such that K is
the largest integer not exceeding (ba)=cn 1=3 ; this roughly equates the spac-
ing of the grid-points to cn 1=3 . Then, the distribution of n 1=3 ( F(t g )F(t g ))
can be approximated by that of the distribution of a random variable that is
characterized as the left-slope of the GCM of a real-valued stochastic process
dened on the grid fcjg j2Z (where Z is the set of integers) and depending
on positive parameters ; that can be consistently estimated from the data.
Section 4 of Tang et al. (2012) provides the details. The random variable that
provides the approximation is easy to generate because the underlying process
that defines it is the restriction of a quadratically drifted Brownian motion
to the grid fcjg j2Z . Tang et al. (2012) demonstrate the effectiveness of their
proposed method through a variety of simulation studies.
As Tang et al. (2012) point out, the underlying principle behind their
\adaptive" method that adjusts to the intrinsic resolution of the grid can be
extended to a variety of settings. Recall that Maathuis and Hudgens (2011)
studied competing risks current status data under grouped or discrete ob-
servation times but did not consider settings where the size of the grid could
depend on the sample size n. As a result, they obtained Gaussian-type asymp-
totics. But again, if the number of discrete observation times is large relative to
the sample size, these Gaussian approximations become unreliable, as demon-
strated in Section 5.1 of their paper. It would therefore be interesting to de-
velop a version of the adaptive procedure in their case, a point noted both
in Section 6 of Maathuis and Hudgens (2011) as well as in Section 6 of Tang