Relevant change points in high dimensional time series

Abstract

This paper investigates the problem of detecting relevant change points in the mean vector, say $\mu_{t}=(\mu_{t,1},\ldots ,\mu_{t,d})^{T}$ 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 \begin{equation*}H_{0}:|\mu^{(1)}_{h}-\mu^{(2)}_{h}|\leq \Delta_{h}~~~\mbox{ forall }~~h=1,\ldots ,d\end{equation*} 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. This formulation of the testing problem is motivated by the fact that in many applications a modification of the statistical analysis might not be necessary, if the differences between the parameters before and after the change points in the individual components are small. This problem is of particular relevance in high dimensional change point analysis, where a small change in only one component can yield a rejection by the classical procedure although all components change only in a non-relevant way.

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 standardized version of the maximum converges weakly to a Gumbel distribution. This result is used to construct a consistent asymptotic level $\alpha $ test and a multiplier bootstrap procedure is proposed, which improves the finite sample performance of the test. The finite sample properties of the test are investigated by means of a simulation study and we also illustrate the new approach investigating data from hydrology.