Higher-order approximation for uncertainty quantification in time series analysis

TL;DR

This paper develops higher-order approximation methods for the empirical process in time series analysis, improving uncertainty quantification and confidence interval accuracy especially for small samples.

Contribution

It introduces a novel approach to calculate confidence intervals using the asymptotic distribution of higher-order error terms in the empirical process.

Findings

Higher-order approximations improve confidence interval coverage.

Simulation shows better accuracy over traditional asymptotic methods.

Method benefits small sample size inference.

Abstract

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. As a result, inference based on the asymptotic distribution of those quantities is significantly affected 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 rates and lengths of these confidence intervals with those based on the asymptotic distribution of the empirical process and highlight some benefits of higher-order approximations of the empirical process.