Biomedical Engineering Reference
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structure in terms of the ordered distinct observation times for the n individ-
uals. Techniques similar to the previous section can therefore be invoked to
study the behavior of the pseudo-LRS. Theorem 1 of Sen and Banerjee (2007)
shows that
n
o
!
d
2
(t
0
)
l
p
n
(
^
n
j X) l
p
n
(
^
0
n
j X)
(t
0
)
D:
The above result provides an easy way of constructing a likelihood ratio-based
2
confidence set for F(t
0
) in the mixed-case interval-censoring model. This is
based on the observation that under the mixed-case interval-censoring frame-
work, where the counting process N(t) is 1(S t) with S following F indepen-
dently of (K;T), the pseudo-likelihood ratio statistic in the above display con-
verges to (1
0
)Dunder the null hypothesis F(t
0
) =
0
. Thus, an asymptotic
level (1 ) condence set for F(t
0
) is f : (1 ) PLRS
n
() q(D; 1 )g,
where q(D; 1) is the (1)-th quantile ofDand PLRS
n
() is the pseudo-
likelihood ratio statistic computed under the null hypothesis H
0;
: F(t
0
) = .
Once again, nuisance parameter estimation has been avoided. An alternative
confidence interval could be constructed by considering the asymptotic dis-
tribution of n
1=3
( F
n;pseudo
(t
0
) F(t
0
)), where F
n;pseudo
is the pseudo-MLE
of F, but this involves a very hard to estimate nuisance parameter; see the
remarks following Theorem 4.4 in Wellner and Zhang (2000).
Relying, as it does, on the marginal likelihoods, the pseudo-likelihood ap-
proach ignores the dependence among the counts at different times points.
An alternative approach is based on considering the full likelihood for a non-
homogeneous Poisson process as studied in Section 2 of Wellner and Zhang
(2000). The MLE of based on the full likelihood was characterized in this
chapter; owing to the lack of separation of variables in the full Poisson like-
lihood (similar to the true likelihood for mixed-case interval-censoring), the
optimization of the likelihood function as well as its analytical treatment are
considerably more complicated. In particular, the analytical behavior of the
MLE of based on the full likelihood does not seem to be known. Wellner
and Zhang (2000) prove an asymptotic result for a \toy" estimator obtained
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