### Abstract

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.

Original language | English |
---|---|

Publisher | ArXiv |

Pages | 1-32 |

Number of pages | 32 |

Publication status | Submitted - 13 Nov 2016 |

### Fingerprint

### Keywords

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

### Cite this

*Tests for scale changes based on pairwise differences*. (pp. 1-32). ArXiv.

**Tests for scale changes based on pairwise differences.** / Gerstenberger, Carina; Vogel, Daniel; Wendler, Martin.

Research output: Working paper

}

TY - UNPB

T1 - Tests for scale changes based on pairwise differences

AU - Gerstenberger, Carina

AU - Vogel, Daniel

AU - Wendler, Martin

N1 - Acknowledgement 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.

PY - 2016/11/13

Y1 - 2016/11/13

N2 - 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.

AB - 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.

KW - Asymptotic relative efficiency

KW - Change-point analysis

KW - Gini’s mean difference

KW - Long-run variance estimation

KW - Median absolute deviation

KW - Qn scale estimator

KW - U-quantile

KW - U-statistic

M3 - Working paper

SP - 1

EP - 32

BT - Tests for scale changes based on pairwise differences

PB - ArXiv

ER -