Testing and Estimating Change-Points in the Covariance Matrix of a High-Dimensional Time Series

TL;DR

This paper develops methods for detecting and estimating change-points in the covariance matrix of high-dimensional time series, applicable to various multivariate models, with theoretical guarantees and practical validation.

Contribution

It introduces a new approach using weighted CUSUM statistics for change-point detection in high-dimensional covariance matrices, with consistent estimators and finite sample analysis.

Findings

Proposed methods effectively detect change-points in simulations.

Provided consistent estimators for unknown variance and covariance parameters.

Applied techniques successfully to real environmental data.

Abstract

This paper studies methods for testing and estimating change-points in the covariance structure of a high-dimensional linear time series. The assumed framework allows for a large class of multivariate linear processes (including vector autoregressive moving average (VARMA) models) of growing dimension and spiked covariance models. The approach uses bilinear forms of the centered or non-centered sample variance-covariance matrix. Change-point testing and estimation are based on maximally selected weighted cumulated sum (CUSUM) statistics. Large sample approximations under a change-point regime are provided including a multivariate CUSUM transform of increasing dimension. For the unknown asymptotic variance and covariance parameters associated to (pairs of) CUSUM statistics we propose consistent estimators. Based on weak laws of large numbers for their sequential versions, we also consider stopped sample estimation where observations until the estimated change-point are used. Finite sample properties of the procedures are investigated by simulations and their application is illustrated by analyzing a real data set from environmetrics.