  In this paper, we study the nonparametric maximum likelihood estimator for an
event time distribution function at a point in the current status model with
observation times supported on a grid of potentially unknown sparsity and with
multiple subjects sharing the same observation time. This is of interest since
observation time ties occur frequently with current status data. The grid
resolution is specified as $cn^{-\gamma}$ with $c>0$ being a scaling constant
and $\gamma>0$ regulating the sparsity of the grid relative to $n$, the number
of subjects. The asymptotic behavior falls into three cases depending on
$\gamma$: regular Gaussian-type asymptotics obtain for $\gamma<1/3$,
nonstandard cube-root asymptotics prevail when $\gamma>1/3$ and $\gamma=1/3$
serves as a boundary at which the transition happens. The limit distribution at
the boundary is different from either of the previous cases and converges
weakly to those obtained with $\gamma\in(0,1/3)$ and $\gamma\in(1/3,\infty)$ as
$c$ goes to $\infty$ and 0, respectively. This weak convergence allows us to
develop an adaptive procedure to construct confidence intervals for the value
of the event time distribution at a point of interest without needing to know
or estimate $\gamma$, which is of enormous advantage from the perspective of
inference. A simulation study of the adaptive procedure is presented.
