  Statistical inference in competing risks models is often based on the famous
Aalen-Johansen estimator. Since the corresponding limit process lacks
independent increments, it is typically applied together with Lin's (1997)
resampling technique involving standard normal multipliers. Recently, it has
been seen that this approach can be interpreted as a wild bootstrap technique
and that other multipliers, as e.g. centered Poissons, may lead to better
finite sample performances, see Beyersmann et al. (2013). Since the latter is
closely related to Efron's classical bootstrap, the question arises whether
this or more general weighted bootstrap versions of Aalen-Johansen processes
lead to valid results. Here we analyze their asymptotic behaviour and it turns
out that such weighted bootstrap versions in general possess the wrong
covariance structure in the limit. However, we explain that the weighted
bootstrap can nevertheless be applied for specific null hypotheses of interest
and also discuss its limitations for statistical inference. To this end, we
introduce different consistent weighted bootstrap tests for the null hypothesis
of stochastically ordered cumulative incidence functions and compare their
finite sample performance in a simulation study.
