Tests for Scale Changes Based on Pairwise Differences

Abstract

In many applications it is important to know whether the amount of fluctuation in a series of observations changes over time. In this article, we investigate different tests for detecting changes in the scale of mean-stationary time series. The classical approach, based on the CUSUM test applied to the squared centered observations, is very vulnerable to outliers and impractical for heavy-tailed data, which leads us to contemplate test statistics based on alternative, less outlier-sensitive scale estimators. It turns out that the tests based on Gini’s mean difference (the average of all pairwise distances) and generalized Q_n estimators (sample quantiles of all pairwise distances) are very suitable candidates. They improve upon the classical test not only under heavy tails or in the presence of outliers, but also under normality. We use recent results on the process convergence of U-statistics and U-quantiles for dependent sequences to derive the limiting distribution of the test statistics and propose estimators for the long-run variance. We show the consistency of the tests and demonstrate the applicability of the new change-point detection methods at two real-life data examples from hydrology and finance. Supplementary materials for this article are available online.

Keywords:

• Block bootstrap
• Gini’s mean difference
• Long-run variance estimation
• U-quantile
• U-statistic

Supplementary Materials

The online supplementary material consists of four files: the two data sets as csv files, the R code of the simulations as R markdown file, and a 13-page appendix file. The appendix file contains three Sections A, B and C. Appendix A refers to Section 2.1 and contains further information on the concept of near-epoch dependence in probability (P-NED). Appendix B refers to Section 2.3. We give general and specific expressions for the population value and the asymptotic variance of $Q_n(\alpha )$ and derive the asymptotic variance of the maximum likelihood scale estimator for t distributions. These are required for Figure 1. Appendix C contains the proofs for the results of Section 3. We also provide intuitive derivation for the long-run variance expressions given in Section 2.2.

Acknowledgments

The authors thank Svenja Fischer for the river Rhine discharge dataset, Marco Thiel for the stock exchange dataset, and Silke Henkes, whose knowledge of Linux helped the simulations to finish two times faster. Their gratitude is extended to the referees and editors, whose valuable comments greatly improved the article.

Additional information

Funding

The authors were supported by the Collaborative Research Centre 823 Statistical modelling of nonlinear dynamic processes and the Konrad-Adenauer-Stiftung.