{"ID":2876717,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.21520","arxiv_id":"2508.21520","title":"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 $δ$ of the mean vectors before and after the change point is equal to zero, we argue that the consideration of the null hypothesis $H_0:\\|δ\\|\\leΔ$, for some norm $\\|\\cdot\\|$ and a threshold $Δ\u003e0$, 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 $δ$. 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 $X_1^\\topξ+\\cdots +X_{\\lfloorλn\\rfloor}^\\topξ$ for a time series $(X_j)_{j\\in\\mathbb{Z}}$ and a contrast vector $ξ\\in\\mathbb{R}^p$ 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.","short_abstract":"We consider the change point testing problem for high-dimensional time series. Unlike conventional approaches, where one tests whether the difference $δ$ of the mean vectors before and after the change point is equal to zero, we argue that the consideration of the null hypothesis $H_0:\\|δ\\|\\leΔ$, for some norm $\\|\\cdot...","url_abs":"https://arxiv.org/abs/2508.21520","url_pdf":"https://arxiv.org/pdf/2508.21520v1","authors":"[\"Pascal Quanz\",\"Holger Dette\"]","published":"2025-08-29T11:13:12Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"math.PR\",\"stat.ME\"]","methods":"[]","has_code":false}