A Test for Trend Gradual Changes in Heavy Tailed AR (p) Sequences

The trend change point is the point at which the trend (or slope) in time series data changes. How to detect such change point is one of the key issues in statistical analysis. This paper proposes for a new gradual change point model for time series trend terms based on the existing abrupt change point model. Secondly, inspired by existing studies, a ratio statistic is constructed for the gradual trend change point in heavy–tailed AR(p) series. The theoretical results indicate that the asymptotic distribution of the statistic under the null hypothesis is a functional of the Lévy process. Meanwhile, this paper proves its consistency under the alternative hypothesis. In addition, due to the heavy tailed characteristics of the sequence, in order to avoid estimating the tail index and reduce the impact of extreme values on the critical values of the statistic, this paper reconstructs the test statistic based on the subsampling method and compares it with the original method. It is found that the subsampling method has a significant improvement on the test power when the change point is located later. Finally, the method is applied to the change point problem of Google stock closing price, and the trend change point is successfully detected.