Current Status Data in the Twenty-First Century: Some Interesting Developments - Interval-Censored-Time-to-Event Data: Methods and Applications - page 70

grouped data case) are allowed to change with n, the properties of the estima-
tors can be quite different, a point I return to later. Maathuis and Hudgens
(2011) also discuss the construction of pointwise confidence intervals for the
sub-distribution functions in the discrete and grouped models as well as the
smooth model. They articulate several diculties with using the limit distri-
bution of the MLEs in the smooth model for setting confidence intervals, such
as nuisance parameter estimation as well as the lack of scaling properties of
the limit. The usual n out of n or model-based bootstrap are both suspect,
although the m out of n bootstrap as well as subsampling can be expected to
work. Maathuis and Hudgens (2011) suggest using inversion of the (pseudo)
likelihood ratio statistic for testing F 0i (t 0 ) = using the pseudo-likelihood
function in Equation (3.6). This is based on the naive estimator and its con-
strained version under the null hypothesis. The likelihood ratio statistic can
be shown to converge toDunder the null hypothesis by methods similar to
Banerjee and Wellner (2001). The computational simplicity of this procedure
makes it attractive even though, owing to the ineciency of the naive esti-
mator with respect to the MLE, these inversion-based intervals will certainly
not be optimal in terms of length. The behavior of the true likelihood ratio
statistic in the smooth model for testing the value of a sub-distribution func-
tion at a point remains completely unknown, and it is unclear whether it will
be asymptotically pivotal.
More recently, Werren (2011) has extended the results of Sen and Baner-
jee (2007) to mixed-case interval-censored data with competing risks. She
defines a naive pseudo-likelihood estimator for the sub-distribution functions
corresponding to the various risks using a working Poisson-process (pseudo)
likelihood, proves consistency, derives the asymptotic limit distribution of the
naive estimators, and presents a method to construct pointwise confidence
intervals for these sub-distribution functions using a pseudo-likelihood ratio
statistic in the spirit of Sen and Banerjee (2007).
I end this section with a brief note on the current status continuous marks