Biomedical Engineering Reference
In-Depth Information
Our approach was to think of mixed-case interval-censored data as data on a
one-jump counting process with counts available only at the inspection times
and to use a pseudo-likelihood function based on the marginal likelihood of a
Poisson process to construct a pseudo-likelihood ratio statistic for testing null
hypotheses of the form H
0
: F(t
0
) =
0
. We showed that under such a null
hypothesis, the statistic converges to a pivotal quantity. Our method was based
on an estimator originally proposed by Sun and Kalbfleisch (1995), whose
asymptotic properties, under appropriate regularity conditions, were studied
in Wellner and Zhang (2000). Indeed, our point of view, that the Interval-
censoring situation can be thought of as a one-jump counting process to which,
consequently, the results on the pseudo-likelihood based estimators can be
applied, was motivated by the latter work. The pseudo-likelihood method
starts by pretending that N(t), the counting process introduced above, is a
nonhomogeneous Poisson process. Then the marginal distribution of N(t) is
given by prfN(t) = kg = expf(t)g(t)
k
=k! for nonnegative integers k.
Note that, under the Poisson process assumption, the successive counts on an
individual (N
K;1
;N
K;2
;:::), conditional on the T
K;j
's, are actually dependent.
However, we ignore the dependence in writing down a likelihood function
for the data. Letting fN
K
i
;T
K
i
;K
i
g
i=1
denote our data X, our likelihood
function, conditional on the T
K
i
's and K
i
's (whose distributions do not involve
), is
K
i
;j
)
N
(i)
K
Y
K
i
;j
)g
(T
(i)
Y
K
i
;j
expf(T
(i)
L
p
n
( j X) =
;
N
(i)
K
i
;j
!
i=1
j=1
and the corresponding log-likelihood up to an irrelevant additive constant is
n
o
X
K
X
N
(i)
K
i
;j
log (T
(i)
K
i
;j
) (T
(i)
l
p
n
( j
X
) =
K
i
;j
)
:
i=1
j=1
Denote by
^
n
and
^
n
, respectively, the unconstrained and constrained
pseudo-MLEs of , with the latter MLE computed under the constraint
(t
0
) =
0
. As is increasing, isotonic estimation techniques apply; fur-
thermore, it is easily seen that the log-likelihood has an additive separated
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