  For time series with high temporal correlation, the empirical process
converges rather slowly to its limiting distribution. Many statistics in
change-point analysis, goodness-of-fit testing and uncertainty quantification
admit a representation as functionals of the empirical process and therefore
inherit its slow convergence. Inference based on the asymptotic distribution of
those quantities becomes highly impacted by relatively small sample sizes. We
assess the quality of higher-order approximations of the empirical process by
deriving the asymptotic distribution of the corresponding error terms. Based on
the limiting distribution of the higher-order terms, we propose a novel
approach to calculate confidence regions for statistical quantities such as the
median. In a simulation study, we compare coverage rate and size of our
confidence regions with those based on the asymptotic distribution of the
empirical process and highlight some of our method's benefits.
