Tests for Scale Changes Based on Pairwise Differences

Carina Gerstenberger, Daniel Vogel* (Corresponding Author), Martin Wendler

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)
3 Downloads (Pure)


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 Qn 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.
Original languageEnglish
Pages (from-to)1336-1348
Number of pages13
JournalJournal of the American Statistical Association
Issue number531
Early online date21 Aug 2019
Publication statusPublished - Sept 2020

Bibliographical note

The researcher were supported by the Collaborative Research Centre 823
Statistical modelling of nonlinear dynamic processes and the Konrad-Adenauer-Stiftung. The authors thank Svenja Fischer for providing the river Rhine discharge data set and Marco Thiel for the stock exchange data set.


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


Dive into the research topics of 'Tests for Scale Changes Based on Pairwise Differences'. Together they form a unique fingerprint.

Cite this