Testing for Change-Points in Heavy-Tailed Time Series—A Winsorized CUSUM Approach

It is well-known that the detection of change-points in heavy-tailed time series is an open problem since the traditional tests may not have a power. This article introduces a winsorized cumulative sum (CUSUM) approach to solve this problem. We begin by investigating the winsorized CUSUM process and then use it to construct the Kolmogorov-Smirnov (KS) test and the Self-normalized (SN) test. Under the null hypothesis, it is shown that each weakly converges to the maximum of a function related to the standard Brownian bridge. Under the alternative, we first study the behavior of tests after applying the winsorization technique, and then show that our tests have a power approaching to 1 as the sample size (Formula presented.). Furthermore, we extend the winsorizing technique to test for multiple change-points without prior knowledge of the number of change points. Our framework is general and its assumptions are mild, so that our tests can be applied to a wide range of linear and nonlinear time series. The empirical results illustrate the effectiveness of our proposed procedures for change-point detection.