Tests for scale changes based on pairwise differences

TL;DR

This paper develops robust statistical tests for detecting scale changes in time series, especially under heavy tails or outliers, by using pairwise difference-based estimators like Gini's mean difference and Qn.

Contribution

It introduces new change-point detection tests based on less outlier-sensitive scale estimators, improving robustness over classical methods, with theoretical derivation and practical validation.

Findings

Tests outperform classical methods under heavy tails and outliers.

Proposed estimators have better long-run variance properties.

Effective in real-world hydrology and finance data.

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 change in the scale of mean-stationary time series. The classical approach based on the CUSUM test applied to the squared centered, 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) or 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. An explanation for this at first counterintuitive result is that the corresponding long-run variance estimates are less affected by a scale change than in the case of the sample-variance-based test. 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 perform a simulations study to investigate the finite sample behavior of the test and their power. Furthermore, we demonstrate the applicability of the new change-point detection methods at two real-life data examples from hydrology and finance.