Practically significant change points in high dimension -- measuring signal strength pro active component

Abstract

We consider the change point testing problem for high-dimensional time series. Unlike conventional approaches, where one tests whether the difference $\delta$ of the mean vectors before and after the change point is equal to zero, we argue that the consideration of the null hypothesis $H_0:\|\delta\|\le\Delta$, for some norm $\|\cdot\|$ and a threshold $\Delta>0$, is better suited. By the formulation of the null hypothesis as a composite hypothesis, the change point testing problem becomes significantly more challenging. We develop pivotal inference for testing hypotheses of this type in the setting of high-dimensional time series, first, measuring deviations from the null vector by the $\ell_2$-norm $\|\cdot\|_2$ normalized by the dimension. Second, by measuring deviations using a sparsity adjusted $\ell_2$-"norm" $\|\cdot \|_2/\sqrt{\|\cdot\|_0} $, where $\|\cdot\|_0$ denotes the $\ell_0$-"norm," we propose a pivotal test procedure which intrinsically adapts to sparse alternatives in a data-driven way by pivotally estimating the set of nonzero entries of the vector $\delta$. To establish the statistical validity of our approach, we derive tail bounds of certain classes of distributions that frequently appear as limiting distributions of self-normalized statistics. As a theoretical foundation for all results, we develop a general weak invariance principle for the partial sum process $X_1^\top\xi +\cdots +X_{\lfloor\lambda n\rfloor}^\top\xi$ for a time series $(X_j)_{j\in\mathbb{Z}}$ and a contrast vector $\xi\in\mathbb{R}^p$ under increasing dimension $p$, which is of independent interest. Finally, we investigate the finite sample properties of the tests by means of a simulation study and illustrate its application in a data example.