Current Status Data in the Twenty-First Century: Some Interesting Developments - Interval-Censored-Time-to-Event Data: Methods and Applications - page 50

petitors is available in Chapter 1 of Banerjee (2000). In the current status
model, the LRS relevant to constructing confidence sets for F(t 0 ) would test
H 0 : F(t 0 ) = 0 against its complement. Is there an asymptotic distribution
for the LRS in this problem? Is the distribution parameter-free? In particular,
is it 2 ? As far as the last query is concerned, the 2 distribution for likelihood
ratios is connected to the differentiability of the finite-dimensional parameter
in the sense of van der Vaart (1991); however, F(t 0 ) is not a differentiable
functional in the interval-censoring model. But even if the limit distribution
of the LRS (if one exists) is different, this does not preclude the possibility
of it being parameter-free. Indeed, this is precisely what Jon Wellner and I
established in Banerjee and Wellner (2001). We found that a particular func-
tional of W(t) + t 2 , which we callD(and which is therefore parameter-free),
describes the large sample behavior of the LRS in the current status model.
This asymptotic pivot can therefore be used to construct confidence sets for
F(t 0 ) by inversion. In subsequent work, I was able to show that the distri-
butionDis a \non-standard" or \nonregular" analogue of the 1 distribution
in nonparametric monotone function estimation problems and can be used
to construct pointwise confidence intervals for monotone functions (via likeli-
hood ratio-based inversion) in a broad class of problems; see Banerjee (2000),
Banerjee and Wellner (2001), Banerjee and Wellner (2005), Sen and Banerjee
(2007), Banerjee (2007), and Banerjee (2009) for some of the important results
along these lines. The first three references deal with the current status model
in detail; the fourth, to which I return later, provides inference strategies for
more general forms of interval-censoring; and the last two deal with extensions
to general monotone function models.
Let me now dwell briefly on the LRS for testing F(t 0 ) = 0 in the current
status model. The log-likelihood function for the observed data f i ;U i g i=1 ,