  We present a method for calculating the full counting statistics of a
nonequilibrium quantum system based on the propagator noncrossing approximation
(NCA). This numerically inexpensive method can provide higher order cumulants
for extended parameter regimes, rendering it attractive for a wide variety of
purposes. We compare NCA results to Born-Markov quantum master equations (QME)
results to show that they can access different physics, and to numerically
exact inchworm quantum Monte-Carlo data to assess their validity. As a
demonstration of its power, the NCA method is employed to study the impact of
correlations on higher order cumulants in the nonequilibrium Anderson impurity
model. The four lowest order cumulants are examined, allowing us to establish
that correlation effects have a profound influence on the underlying transport
distributions. Higher order cumulants are therefore demonstrated to be a proxy
for the presence of Kondo correlations in a way that cannot be captured by
simple QME methods.
