Current Status Data in the Twenty-First Century: Some Interesting Developments - Interval-Censored-Time-to-Event Data: Methods and Applications - page 61

(1996) shows that n 1=3 ( F n (t 0 )F(t 0 )) converges in distribution to a constant
timesZ.
It is, then, natural to be curious as to whether the LRS for testing F(t 0 ) =
0 is again asymptotically characterized byD. Unfortunately, this has still not
been established. One key reason behind this is the fact that the computation
of the constrained MLE of F under the hypothesis F(t 0 ) = 0 can no longer
be decomposed into two separate optimization problems, in contrast to the
current status model in the previous section or the monotone response models
in Banerjee (2007). A self-induced characterization of the constrained NPMLE
is still available but computationally more dicult to implement. Furthermore,
the techniques for separated monotone function models that enable us to get a
handle on the relationships between the unconstrained and constrained MLEs
of F and, in particular the set on which they differ (which plays a crucial role
in studying the LRS) do not seem to work either. Nevertheless, some rough
heuristics (which involve some conjectures about the relation of F n to F n )
indicate thatDmay, yet again, be the distributional limit of the LRS. As a
first step, one would want to implement a progam to compute the LRS in the
Case 2 model and check whether there is empirical agreement between the
quantiles from its distribution and the quantiles ofD.
The complexities with the Case 2 model are of course present with mixed-
case censoring. Song (2004) studied estimation with mixed-case interval-
censored data, characterized and computed the NPMLE for this model, and
established asymptotic properties like consistency, global rates of convergence,
and an asymptotic minimax lower bound but does not have a pointwise limit
distribution result analogous to that in Groeneboom (1996).
The question, then, is whether one can postulate an asymptotically pivotal
method (as in the current status case) for estimation of F(t 0 ) in the mixed-case
model. Fortunately, Bodhi Sen and I were able to provide a positive answer to
this question in Sen and Banerjee (2007) by treating the mixed-case model as a
special case of the panel data model introduced at the beginning of this section.