[Paper Review] Dating the Break in High-dimensional Data

Runmin Wang, Xiaofeng Shao|arXiv (Cornell University)|Feb 10, 2020

Statistical Methods and Inference16 references5 citations

TL;DR

This paper proposes a novel U-statistic-based estimator for locating a change point in the mean of high-dimensional, independent data, achieving superior efficiency over least squares methods. The method establishes asymptotic normality and confidence intervals via asymptotic theory and bootstrap resampling, valid even when p >> n and components are dependent.

ABSTRACT

This paper is concerned with estimation and inference for the location of a change point in the mean of independent high-dimensional data. Our change point location estimator maximizes a new U-statistic based objective function, and its convergence rate and asymptotic distribution after suitable centering and normalization are obtained under mild assumptions. Our estimator turns out to have better efficiency as compared to the least squares based counterpart in the literature. Based on the asymptotic theory, we construct a confidence interval by plugging in consistent estimates of several quantities in the normalization. We also provide a bootstrap-based confidence interval and state its asymptotic validity under suitable conditions. Through simulation studies, we demonstrate favorable finite sample performance of the new change point location estimator as compared to its least squares based counterpart, and our bootstrap-based confidence intervals, as compared to several existing competitors. The asymptotic theory based on high-dimensional U-statistic is substantially different from those developed in the literature and is of independent interest.

Motivation & Objective

• Address the challenge of detecting structural breaks in high-dimensional data where p >> n and components may be dependent.
• Develop a new estimation method for change point location that outperforms existing least squares-based approaches in efficiency.
• Establish asymptotic theory for the proposed estimator under mild regularity conditions, including convergence rate and asymptotic distribution.
• Construct both asymptotic and bootstrap-based confidence intervals for the change point location, ensuring validity under high-dimensional settings.
• Provide theoretical justification and finite-sample validation for the proposed methods in complex, high-dimensional data structures.

Proposed method

• Propose a new U-statistic-based objective function to maximize for change point estimation, replacing traditional least squares.
• Derive the convergence rate and asymptotic distribution of the maximizer of the U-statistic under mild moment and dependence assumptions.
• Use consistent estimators of normalization constants to construct asymptotic confidence intervals for the change point location.
• Introduce a bootstrap-based confidence interval procedure that adapts to the magnitude of change and is theoretically justified under weak dependence.
• Leverage high-dimensional U-statistic theory to derive limit distributions, differing significantly from classical low-dimensional asymptotic frameworks.
• Apply Hájek-Rényi and Kolmogorov-type inequalities to control tail probabilities of partial sums and cross-products in high-dimensional settings.

Experimental results

Research questions

• RQ1 Can a U-statistic-based estimator achieve better efficiency than least squares in high-dimensional change point detection?
• RQ2 What is the asymptotic distribution of the change point estimator under high-dimensional and dependent data assumptions?
• RQ3 How can valid confidence intervals be constructed for the change point location when p >> n and components are dependent?
• RQ4 Does the proposed bootstrap-based interval maintain asymptotic validity in high-dimensional settings with unknown change magnitude?
• RQ5 What are the theoretical and finite-sample performance advantages of the new method over existing least squares and CUSUM-based approaches?

Key findings

• The proposed U-statistic-based estimator achieves higher efficiency than the least squares-based estimator in certain high-dimensional models.
• The estimator is consistent and converges at a rate of $ O_p(n^{-1/2}) $, with a non-degenerate asymptotic distribution after proper centering and normalization.
• The asymptotic confidence interval is valid under mild assumptions and performs well in finite samples, especially when the change magnitude is moderate.
• The bootstrap-based confidence interval is asymptotically valid and shows favorable finite-sample performance, adapting to unknown change size.
• Theoretical analysis reveals that high-dimensional U-statistic behavior differs fundamentally from classical low-dimensional theory, particularly in the dependence structure and normalization constants.
• Simulation studies confirm the proposed estimator's superior finite-sample performance compared to least squares and other competitors, particularly in detecting change points under high-dimensional and dependent settings.