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 $n/d\to0$. 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 $\max_{1\leq h\leq d}\sup_{0\leq t\leq1}\vert \mathcal{W}_{t,h}-t\mathcal{W}_{1,h}\vert$ is established, where $\{\mathcal{W}_{t,h}\}_{1\leq h\leq d}$ denotes a sequence of dependent Brownian motions. As an application, we analyze all S&P 500 companies over a period of one year.Comment: Clarified Assumption 4.3 and Theorems 4.4, 4.6 and 4.8. The results themselves are unchange

Relevant change points in high dimensional time series

“…and the claim follows from Theorem 2.5 in (Jirak, 2015a) noting that {U n,h (s) − E [U n,h (s)]} s∈[0,1] corresponds to a CUSUM-process under the classical null hypothesis of no change point, that is µ…”

“…whereσ h denotes an estimator for the unknown long-run variance (see Section 3.3 for a precise definition), and the quantityτ h is a function of the estimate of the change point defined bŷ Note that a similar approach has been investigated by Jirak (2015a), who considered the "classical" change point problem in high dimension, that is…”

“…The following Lemma shows that these estimators are uniformly consistent with respect to all components, where a change point exists. Its proof follows by an adaption of Corollary 3.1 in Jirak (2015a), and is therefore omitted.…”

High Dimensional Change Point Estimation via Sparse Projection

“…We compare the performance of the inspect algorithm with the following recently proposed methods for high dimensional change point estimation: the sparsified binary segmentation algorithm sbs (Cho and Fryzlewicz, ), the double‐CUSUM algorithm dc of Cho (), the scan‐statistic‐based algorithm scan derived from the work of Enikeeva and Harchaoui (), the CUSUM aggregation algorithm of Jirak () and the l 2 CUSUM aggregation algorithm agg 2 of Horváth and Hušková (). We remark that the latter three works primarily concern the test for the existence of a change point.…”

“…Enikeeva and Harchaoui () also considered a scan statistic that takes sparsity into account. Jirak () considered an ‐aggregation of the CUSUM statistics that works well for sparse change points. Cho and Fryzlewicz () proposed sparse binary segmentation, which also takes sparsity into account and can be viewed as a hard thresholding of the CUSUM matrix followed by an l 1 ‐aggregation.…”