  We study the uniform convergence rate of the nonparametric maximum likelihood
estimator (MLE) for the sub-distribution functions in the current status data
with competing risks model. It is known that the MLE have $L^2$-norm
convergence rate $O_P(n^{-1/3})$ in the absolutely continuous case, but there
is no arguments for the same rate of uniform convergence. We specify conditions
for the uniform convergence rate $O_P(n^{-1/3}\log^{1/3} n)$ of the MLE for the
sub-distribution functions of competing risks on finite intervals. The obtained
result refines known uniform convergence rate in the particular case of current
status data. The main result is applied in order to get the uniform convergence
rate of the MLE for the survival function of failure time in the current status
right-censored data model.
