Tests for Scale Changes Based on Pairwise Differences

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

Research output: Contribution to journalArticle


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
JournalJournal of the American Statistical Association
Early online date21 Aug 2019
Publication statusE-pub ahead of print - 21 Aug 2019



  • Asymptotic relative efficiency
  • Change-point analysis
  • Gini’s mean difference
  • Long-run variance estimation
  • Median absolute deviation
  • Qn scale estimator
  • U-quantile
  • U-statistic
  • Block bootstrap
  • Gini's mean difference
  • SUMS

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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