Detecting scale changes using pairwise differences

The mean of all pairwise differences is commonly referred to as Gini’s mean difference. Rousseeuw & Croux (1993) propose to use the 1/4th sample quantile of all pairwise differences and call the resulting estimator Qn . Both estimators are popular scale estimators that combine very good statistical properties with an intriguing conceptual simplicity. We consider these estimators in the context of change-point analysis. We review their efficiency and robustness properties and then construct CUSUM-type change-point test statistics based on Gini’s mean difference and various sample quantiles of the pairwise differences. We use recent results on the asymptotics of U-statistics and U-quantiles for dependent data to derive critical values. The behavior of the tests is examined by means of numerical simulations, demonstrating their general superiority over the classical second-moments-based CUSUM test for detecting scale changes.

[joint work with C. Gerstenberger and M. Wendler]

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