Current Status Data in the Twenty-First Century: Some Interesting Developments - Interval-Censored-Time-to-Event Data: Methods and Applications - page 66

somewhat involved; as Jewell and van der Laan (2003) note, the general EM

algorithm can be used for this purpose but is extremely slow. Jewell and
Kalbfleisch (2004) developed a much faster iterative algorithm that generalizes

the PAVA; the pooling now involves solving a polynomial equation instead of

simple averaging, the latter being the case with standard current status data.
We denote the MLE of (F 1 ;F 2 ) by ( F 1 ; F 2 ). A competing estimator is the
so-called \naive estimator," which was also studied in Jewell et al. (2003), and
we denote this by ( F 1 ; F 2 ). Here, F i = max F L ni (F), where F is a generic
sub-distribution function and

Y

F(U k ) i (1 F(U k )) 1 i :

L ni (F) = (3.6)

k=1

Thus, the naive estimator separates the estimation problem into two sepa-

rate well-known univariate current status problems, and the properties of the

naive estimator follow from the same arguments that work in the simple cur-

rent status model. The problem, however, lies in that by treating 1 and 2
separately, a critical feature of the data is ignored and the natural estimate
of F + , F + = F 1 + F 2 , may no longer be a proper distribution function (it can
be larger than 1). Both the MLE and the naive estimator are consistent but

the MLE turns out to be more ecient than the naive estimator, as we will

see below. Groeneboom et al. (2008a) and Groeneboom et al. (2008b) develop

the full asymptotic theory for the MLE and the naive estimators. The naive
estimator, of course, converges pointwise at rate n 1=3 but figuring out the local

rates of convergence of the MLEs of the sub-distribution functions takes a lot

of work. As Groeneboom et al. (2008a) note in their introduction, the proof
of the local rate of convergence of F 1 and F 2 requires new ideas that go well
beyond those needed for the simple current status model or general monotone

function models. One of the major diculties in the proof lies in the delicate

handling of the system of sub-distribution functions.
This requires an initial result on the convergence rate of F + uniformly
on a fixed neighborhood of t 0 , and is accomplished in Theorem 4.10 of their