# Inference for Change Points in High Dimensional Data via Self-Normalization

## TL;DR

This paper introduces a new self-normalized testing method for detecting change points in high-dimensional data, applicable to both independent and time series data, with theoretical guarantees and practical estimation procedures.

## Contribution

It develops a novel self-normalized test for high-dimensional change points that requires no tuning parameters and extends to dependent data with a trimming approach.

## Key findings

- The proposed tests are theoretically justified under null and alternative hypotheses.
- Numerical simulations show the methods outperform existing approaches.
- The approach can accurately estimate multiple change points using wild binary segmentation.

## Abstract

This article considers change point testing and estimation for a sequence of high-dimensional data. In the case of testing for a mean shift for high-dimensional independent data, we propose a new test which is based on $U$-statistic in Chen and Qin (2010) and utilizes the self-normalization principle [Shao (2010), Shao and Zhang (2010)]. Our test targets dense alternatives in the high-dimensional setting and involves no tuning parameters. To extend to change point testing for high-dimensional time series, we introduce a trimming parameter and formulate a self-normalized test statistic with trimming to accommodate the weak temporal dependence. On the theory front, we derive the limiting distributions of self-normalized test statistics under both the null and alternatives for both independent and dependent high-dimensional data. At the core of our asymptotic theory, we obtain weak convergence of a sequential U-statistic based process for high-dimensional independent data, and weak convergence of sequential trimmed U-statistic based processes for high-dimensional linear processes, both of which are of independent interests. Additionally, we illustrate how our tests can be used in combination with wild binary segmentation to estimate the number and location of multiple change points. Numerical simulations demonstrate the competitiveness of our proposed testing and estimation procedures in comparison with several existing methods in the literature.