Optimal detection of general moment changes: Simultaneous mean and covariance change detection and beyond

2026-09-28

AI Summary

This study addresses the dual challenges of growing dimensionality and temporal dependence in detecting changes across all moments of high-dimensional multivariate time series. The proposed method leverages a linear algebraic framework with tensor representations to uniformly characterize moments of arbitrary order, enabling the simultaneous detection of change points for any fixed-order moment. By integrating asymptotic statistical theory, the authors derive optimal localization error rates and construct asymptotically valid confidence intervals. The resulting estimation accuracy achieves the minimax lower bound, demonstrating theoretical optimality. Extensive numerical experiments and real-data applications further validate the robustness of this approach in detecting diverse distributional shifts.

Abstract

We study multiple change-point detection in multivariate time series whose distributions change in a piecewise constant manner. Distributional changes can manifest across different moment orders, from shifts in the mean and covariance to changes in higher-order moments. Higher-order moments capture increasingly rich distributional features but become difficult to estimate in high dimensions. Our tensor representation unifies moments of different orders within a common linear algebraic framework, enabling a new method to detect changes in moments of all orders up to a prescribed fixed order $p$. The resulting procedure accommodates temporal dependence and allows the dimension of the time series to grow with the sample size. Under suitable regularity conditions, the proposed procedure achieves a localization error rate that matches a newly developed minimax lower bound. We further derive limiting distributions under both nonvanishing and vanishing moment jumps and construct asymptotically valid confidence intervals in the vanishing-jump regime. Numerical experiments and real-data analyses demonstrate the method's effectiveness in detecting moment changes and a range of distributional shifts.

Problem

• change-point detection
• multivariate time series
• moment changes
• high-dimensional data
• minimax optimality

Innovation

• change-point detection
• tensor representation
• high-dimensional time series
• minimax lower bound
• higher-order moments