TY - JOUR

T1 - Testing for change points in time series

AU - Shao, Xiaofeng

AU - Zhang, Xianyang

N1 - Funding Information:
Xiaofeng Shao is Assistant Professor (E-mail: [email protected]) and Xi-anyang Zhang is Ph.D. Candidate, Department of Statistics, University of Illinois at Urbana–Champaign, Champaign, IL 61820. We are grateful to the two referees and Zhibiao Zhao for detailed comments, which led to substantial improvements. We thank Wei Biao Wu for providing Argentina rainfall data used in this paper. The research is supported in part by NSF grants DMS-0804937 and DMS-0724752.

PY - 2010/9

Y1 - 2010/9

N2 - This article considers the CUSUM-based (cumulative sum) test for a change point in a time series. In the case of testing for a mean shift, the traditional Kolmogorov-Smirnov test statistic involves a consistent long-run variance estimator, which is needed to make the limiting null distribution free of nuisance parameters. The commonly used lag-window type long-run variance estimator requires to choose a bandwidth parameter and its selection is a difficult task in practice. The bandwidth that is a fixed function of the sample size (e.g., n1/3, where n is sample size) is not adaptive to the magnitude of the dependence in the series, whereas the data-dependent bandwidth could lead to nonmonotonic power as shown in previous studies. In this article, we propose a self-normalization (SN) based Kolmogorov-Smirnov test, where the formation of the self-normalizer takes the change point alternative into account. The resulting test statistic is asymptotically distribution free and its power is monotonic. Furthermore, we extend the SN-based test to test for a change in other parameters associated with a time series, such as marginal median, autocorrelation at lag one, and spectrum at certain frequency bands. The use of the SN idea thus allows a unified treatment and offers a new perspective to the large literature of change point detection in the time series setting. Monte Carlo simulations are conducted to compare the finite sample performance of the new SN-based test with the traditional Kolmogorov-Smirnov test. Illustrations using real data examples are presented.

AB - This article considers the CUSUM-based (cumulative sum) test for a change point in a time series. In the case of testing for a mean shift, the traditional Kolmogorov-Smirnov test statistic involves a consistent long-run variance estimator, which is needed to make the limiting null distribution free of nuisance parameters. The commonly used lag-window type long-run variance estimator requires to choose a bandwidth parameter and its selection is a difficult task in practice. The bandwidth that is a fixed function of the sample size (e.g., n1/3, where n is sample size) is not adaptive to the magnitude of the dependence in the series, whereas the data-dependent bandwidth could lead to nonmonotonic power as shown in previous studies. In this article, we propose a self-normalization (SN) based Kolmogorov-Smirnov test, where the formation of the self-normalizer takes the change point alternative into account. The resulting test statistic is asymptotically distribution free and its power is monotonic. Furthermore, we extend the SN-based test to test for a change in other parameters associated with a time series, such as marginal median, autocorrelation at lag one, and spectrum at certain frequency bands. The use of the SN idea thus allows a unified treatment and offers a new perspective to the large literature of change point detection in the time series setting. Monte Carlo simulations are conducted to compare the finite sample performance of the new SN-based test with the traditional Kolmogorov-Smirnov test. Illustrations using real data examples are presented.

KW - CUSUM

KW - Invariance principle

KW - Self-normalization

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U2 - 10.1198/jasa.2010.tm10103

DO - 10.1198/jasa.2010.tm10103

M3 - Article

AN - SCOPUS:78649415337

SN - 0162-1459

VL - 105

SP - 1228

EP - 1240

JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

IS - 491

ER -