Current Status Data in the Twenty-First Century: Some Interesting Developments - Interval-Censored-Time-to-Event Data: Methods and Applications - page 49

selection. The above considerations then raise a natural question: Can we pre-

scribe confidence intervals that obviate the need to estimate these nuisance
parameters? Indeed, this is what set Jon Wellner and me thinking of alter-

native solutions to the problem around 2000. That the usual Efron-type n

out of n bootstrap is unreliable in this situation was already suspected; see,

for example, the introduction of Delgado et al. (2001). While the m out of

n bootstrap or its variant, subsampling, works in this situation, the selection

of m is tricky and analogous to a bandwidth selection problem, which our

goal was to avoid. As it turned out, in this problem, likelihood ratios would

come to the rescue. The possible use of likelihood ratios in the current status

problem was motivated by the then-recent work of Murphy and van der Vaart

(1997) and Murphy and van der Vaart (2000) on likelihood ratio inference for

the finite-dimensional parameter in regular semiparametric models. Murphy

and van der Vaart showed that in semiparametric models, the likelihood ratio

statistic (LRS) for testing H 0 : = 0 against its complement, being a path-
wise norm-differentiable finite-dimensional parameter in the model, converges
under the null hypothesis to a 2 distribution with the number of degrees of

freedom matching the dimensionality of the parameter. This result, which is

analogous to what happens in purely parametric settings, provides a conve-

nient way to construct confidence intervals via the method of inversion: an
asymptotic level 1 condence set is given by the set of all 0 for which the
LRS for testing H 0; 0 : = 0 (against its complement) is no larger than the
(1 )-th quantile of a 2

distribution. This is a clean method as nuisance

parameters need not be estimated from the data; in contrast, the Wald-type

confidence ellipsoids that use the asymptotic distribution of the MLE would

require estimating the information matrix.

Furthermore, likelihood ratio-based condence sets are more \data-driven"

than the Wald-type sets, which necessarily have a prespecified shape and sat-

isfy symmetry properties about the MLE. An informative discussion of the

several advantages of likelihood ratio-based confidence sets over their com-