Estimating a Change Point in a Sequence of Very High-Dimensional Covariance Matrices

Abstract

This article considers the problem of estimating a change point in the covariance matrix in a sequence of high-dimensional vectors, where the dimension is substantially larger than the sample size. A two-stage approach is proposed to efficiently estimate the location of the change point. The first step consists of a reduction of the dimension to identify elements of the covariance matrices corresponding to significant changes. In a second step, we use the components after dimension reduction to determine the position of the change point. Theoretical properties are developed for both steps, and numerical studies are conducted to support the new methodology. Supplementary materials for this article are available online.

Keywords:

• Change point analysis
• Dimension reduction
• High-dimensional covariance matrices

Supplementary Materials

The online supplementary files contain all proofs and technical details.

Acknowledgments

The authors would like to thank M. Stein who typed parts of this article with considerable technical expertise. We are also grateful to V. Avanesov and N. Buzun for making the R-code of their procedure available to us.

Additional information

Funding

The work of H. Dette was partially supported by the Deutsche Forschungsgemeinschaft (DFG Research Unit 1735). G. M. Pan was partially supported by a MOE Tier 2 grant 2018-T2-2-112 and a MOE Tier 1 Grant RG133/18 at the Nanyang Technological University, Singapore.