Adaptive Inference for Change Points in High-Dimensional Data

Abstract

In this article, we propose a class of test statistics for a change point in the mean of high-dimensional independent data. Our test integrates the U-statistic based approach in a recent work by Wang et al. and the L_q-norm based high-dimensional test in a recent work by He et al., and inherits several appealing features such as being tuning parameter free and asymptotic independence for test statistics corresponding to even q’s. A simple combination of test statistics corresponding to several different q’s leads to a test with adaptive power property, that is, it can be powerful against both sparse and dense alternatives. On the estimation front, we obtain the convergence rate of the maximizer of our test statistic standardized by sample size when there is one change-point in mean and q = 2, and propose to combine our tests with a wild binary segmentation algorithm to estimate the change-point number and locations when there are multiple change-points. Numerical comparisons using both simulated and real data demonstrate the advantage of our adaptive test and its corresponding estimation method.

Keywords:

• Asymptotically pivotal
• Segmentation
• Self-normalization
• Structural break
• U-statistics

Supplementary Materials

In the supplement, Section 6 contains all technical proofs for theoretical results stated in the paper and some auxiliary lemmas. Section 7 presents additional simulation results for network change-point testing and estimation.

Acknowledgments

We would like to thank two anonymous referees for constructive comments, which led to substantial improvements. We are also grateful to Dr. Farida Enikeeva for sending us the code used in Enikeeva and Harchaoui (2019).

Additional information

Funding

Shao’s research is partially supported by NSF-DMS 1807023 and NSF-DMS-2014018.