A tail adaptive approach for change point detection

Abstract

For change point problems with Gaussian distributions, the CUSUM method is most efficient for detecting mean shifts. In contrast, it is not so efficient for heavy-tailed or contaminated data because of its sensitivity to outliers. To address this issue, Csörgő and Horváth (1988) introduced the Wilcoxon–Mann–Whitney test based on two-sample $U$-statistics. In practice, however, the tail structure of distributions is typically unknown. For example, Barndorff-Nielsen and Shephard (2001) showed that with higher frequency, stock returns’ tails become heavier. To our knowledge, there are no uniformly most powerful testing methods for both heavy and light-tailed distributions. To deal with this issue, we construct a new family of test statistics and combine them to adapt to different tails. As the final test statistic is complex, we design a low-cost bootstrap procedure to approximate its limiting distribution. To capture temporal data dependence, we assume that the data follow a near epoch dependent process (Borovkova et al., 2001), which includes ARMA and GARCH processes, among others. We explore the validity of our method both theoretically and through simulation. We also illustrate its use with data on the S&P 500 index.