Biomedical Engineering Reference
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model. Let X be an event time and Y a jointly distributed continuous \mark"
variable with joint distribution F
0
. In the current status continuous mark
model, instead of observing (X;Y ), we observe a continuous censoring variable
U, independent of (X;Y ) and the indicator variable = 1fX Ug. If = 1,
we also observe the \mark" variable Y ; in case = 0, the variable Y is not
observed. Note that this is precisely a continuous version of the competing
risks model: the discrete random variable Y in the usual competing risks model
has now been changed to a continuous variable. Maathuis and Wellner (2008)
consider a more general version of this model where, instead of current status
censoring, they have general case k censoring. They derive the MLE of F
0
and
its almost sure limit, which leads to necessary and sucient conditions for
consistency of the MLE. However, these conditions force a relation between
the unknown distribution F
0
and G, the distribution of U. Because such a
relation is typically not satisfied, the MLE is inconsistent in general in the
continuous marks model. Inconsistency of the MLE can be removed by either
discretizing the marks, as in Maathuis and Wellner (2008); an alternative
strategy is to use appropriate kernel smoothed estimators of F
0
(instead of
the MLE), which will be discussed briefly in the next section.
3.5
Smoothed Estimators for Current Status Data
I first deal with the simple current status model that has been the focus
of much of the previous sections using the same notation as in Section 3.2.
While the MLE of F in the current status model does not require bandwidth
specification and achieves the best possible pointwise convergence rate un-
der minimal smoothness (F only needs to be continuously differentiable in a
neighborhood of t
0
, the point of interest), it is not the most optimal estimate
in terms of convergence rates if one is willing to assume stronger smoothness
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