Relevant change points in high dimensional time series

• Published in: arXiv.org 2021-02
• Main Authors: Dette, Holger, Gösmann, Josua
• Format: Article
• Language: English
• Subjects: Change detection; Convergence; Dependent variables; Dimensional changes; Hypotheses; Null hypothesis; Random variables; Time series
• ISSN: 2331-8422

This paper investigates the problem of detecting relevant change points in the mean vector, say \(\mu_t =(\mu_{1,t},\ldots ,\mu_{d,t})^T\) of a high dimensional time series \((Z_t)_{t\in \mathbb{Z}}\). While the recent literature on testing for change points in this context considers hypotheses for the equality of the means \(\mu_h^{(1)}\) and \(\mu_h^{(2)}\) before and after the change points in the different components, we are interested in a null hypothesis of the form $$ H_0: |\mu^{(1)}_{h} - \mu^{(2)}_{h} | \leq \Delta_h ~~~\mbox{ for all } ~~h=1,\ldots ,d $$ where \(\Delta_1, \ldots , \Delta_d\) are given thresholds for which a smaller difference of the means in the \(h\)-th component is considered to be non-relevant. We propose a new test for this problem based on the maximum of squared and integrated CUSUM statistics and investigate its properties as the sample size \(n\) and the dimension \(d\) both converge to infinity. In particular, using Gaussian approximations for the maximum of a large number of dependent random variables, we show that on certain points of the boundary of the null hypothesis a standardised version of the maximum converges weakly to a Gumbel distribution.