Biomedical Engineering Reference
In-Depth Information
of U
i
's and V
i
's, that is, R(U
i
) = s if Ui
i
= t
s
and R(V
i
) = p if Vi
i
= t
p
. Then,
n
(1)
i
X
l
n
=
log F(t
R(U
i
)
)
i=1
+
(2)
i
log(F(t
R(V
i
)
) F(t
R(U
i
)
))
+
(3)
i
log(1 F(t
R(V
i
)
))
o
:
Now, l
n
as a function in (F(t
1
);F(t
2
);:::;F(t
J
)) is concave and thus, finding
the NPMLE of F boils down to maximizing a concave function over a convex
cone as in the current status problem. However, the structure of l
n
is now
considerably more involved than in the current status model. If we go back to
l
n
in the current status model we see that it is the sum of n univariate con-
cave functions with the i-th function involving F(U
(i)
) and the corresponding
response
(i)
; thus we have a separation of variables. With the Case 2 log-
likelihood, this is no longer the case as terms of the form log(F(t
i
) F(t
j
))
enter the likelihood and l
n
no longer has an additive (separated) structure in
the F(t
i
)'s. The nonseparated structure in Case 2 interval-censoring leads to
some complications: first, there is no longer an explicit solution to the NPMLE
via PAVA; rather, F
n
has a self-induced characterization as the slope of the
GCM of a stochastic process that depends on F
n
itself. See, for example, Chap-
ter 2 of Groeneboom and Wellner (1992). The computation of F
n
relies on the
ICM (iterative convex minorant) algorithm that is discussed in Chapter 3 of
the same book and was subsequently modified for effective implementation in
Jongbloed (1998), where the Case 2 log-likelihood was used as a test example.
Second, and more importantly, the nonseparated log-likelihood is quite di-
cult to handle. Groeneboom (1996) had to use some very hard analysis to get
around the lack of separation and establish the pointwise asymptotic distribu-
tion of F
n
in the Case 2 censoring model. Under certain regularity conditions
for which we refer the reader to the original manuscript, the most critical of
which is that V U is larger than some positive number with probability one
(this condition is very natural in practical applications because there is always
a minimal gap between the first and second inspection times), Groeneboom
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