Higher order approximation for constructing confidence intervals in time series

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 intervals for statistical quantities such as the median. In a simulation study, we compare coverage rate and interval length of our confidence intervals with confidence intervals based on the asymptotic distribution of the empirical process and highlight some of the benefits of our method.