Uniform change point tests in high dimension

Abstract

Consider $d$ dependent change point tests, each based on a CUSUM-statistic. We provide an asymptotic theory that allows us to deal with the maximum over all test statistics as both the sample size $n$ and $d$ tend to infinity. We achieve this either by a consistent bootstrap or an appropriate limit distribution. This allows for the construction of simultaneous confidence bands for dependent change point tests, and explicitly allows us to determine the location of the change both in time and coordinates in high-dimensional time series. If the underlying data has sample size greater or equal $n$ for each test, our conditions explicitly allow for the large $d$ small $n$ situation, that is, where $n/d\to 0$. The setup for the high-dimensional time series is based on a general weak dependence concept. The conditions are very flexible and include many popular multivariate linear and nonlinear models from the literature, such as ARMA, GARCH and related models. The construction of the tests is completely nonparametric, difficulties associated with parametric model selection, model fitting and parameter estimation are avoided. Among other things, the limit distribution for ${\max }_{1\le h\le d}{\sup }_{0\le t\le 1}|{\mathcal{W}}_{t,h}-t{\mathcal{W}}_{1,h}|$ is established, where $\{{\mathcal{W}}_{t,h}{\}}_{1\le h\le d}$ denotes a sequence of dependent Brownian motions. As an application, we analyze all S&P 500 companies over a period of one year.

Clarified Assumption 4.3 and Theorems 4.4, 4.6 and 4.8. The results themselves are unchanged

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• High-Dimensional Changepoint Detection via a Geometrically Inspired Mapping
• Optimal Covariance Change Point Localization in High Dimension
• Data segmentation algorithms: Univariate mean change and beyond
• A review on minimax rates in change point detection and localisation
• Univariate Mean Change Point Detection: Penalization, CUSUM and Optimality
• Multiscale Jump Testing and Estimation Under Complex Temporal Dynamics
• Change Detection: A functional analysis perspective
• Dating the Break in High-dimensional Data
• Valid and Exact Statistical Inference for Multi-dimensional Multiple Change-Points by Selective Inference
• Selective Inference for Change Point Detection in Multi-dimensional Sequences
• Asymptotic Distribution-Free Change-Point Detection for Multivariate and non-Euclidean Data
• Test for Temporal Homogeneity of Means in High-dimensional Longitudinal Data
• Adaptive Inference for Change Points in High-Dimensional Data
• Structural break analysis in high-dimensional covariance structure
• Segmentation of high dimensional means over multi-dimensional change points and connections to regression trees
• Estimation of high-dimensional change-points under a group sparsity structure
• Spectral Change Point Estimation for High Dimensional Time Series by Sparse Tensor Decomposition
• Robust change-point detection in panel data
• High-dimensional GARCH process segmentation with an application to Value-at-Risk
• Inference on the change point in high dimensional time series models via plug in least squares