  We study nonparametric estimation for current status data with competing
risks. Our main interest is in the nonparametric maximum likelihood estimator
(MLE), and for comparison we also consider a simpler ``naive estimator.''
Groeneboom, Maathuis and Wellner [Ann. Statist. (2008) 36 1031--1063] proved
that both types of estimators converge globally and locally at rate $n^{1/3}$.
We use these results to derive the local limiting distributions of the
estimators. The limiting distribution of the naive estimator is given by the
slopes of the convex minorants of correlated Brownian motion processes with
parabolic drifts. The limiting distribution of the MLE involves a new
self-induced limiting process. Finally, we present a simulation study showing
that the MLE is superior to the naive estimator in terms of mean squared error,
both for small sample sizes and asymptotically.
