Tests for scale changes based on pairwise differences
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Date
2016
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Abstract
In many applications it is important to know whether the amount of
uctuation 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 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 simulation study to investigate the
finite sample behavior of the tests 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.
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Keywords
asymptotic relative efficiency, U-statistic, U-quantile, Qn scale estimator, median absolute deviation, long-run variance estimation, Gini's mean difference, change-point analysis