Journal Article

Testing for Change Points in Time Series

Zhang, Xianyang, Shao, Xiaofeng

2010

Overview

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., n 1/3 , 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.

• Publisher: Taylor & Francis,American Statistical Association,Assoc,Taylor & Francis Ltd
• Subject: Adaptive sampling / Analytical estimating / Applications / Asymptotic methods / Bandwidths / Changes / Comparative analysis / Consistent estimators / Correlation / Critical values / CUSUM / Data analysis / Electric power distribution / Estimators / Exact sciences and technology / Frequencies / General topics / Inference from stochastic processes; time series analysis / Invariance principle / Limit theorems / Mathematics / Model testing / Monte Carlo simulation / Nonmonotonic logic / Normalization / Parameters / Point estimators / Power / Probability and statistics / Probability theory and stochastic processes / Regression analysis / Sample size / Sciences and techniques of general use / Self-normalization / Statistical data / Statistical methods / Statistical variance / Statistics / Tests / Theory and Methods / Time / Time series / Time series models / Variance