Adaptive Ridge-Regularized Hotelling Change-Point Tests for Functional Data

2026-07-18

AI Summary

This study addresses the detection and localization of mean change points in high-dimensional, weakly dependent, non-Gaussian functional time series. The proposed approach maps functional observations to high-dimensional score vectors via basis expansion, estimates the long-run covariance using an edge-corrected differencing scheme, and stabilizes the Hotelling statistic through adaptive ridge regularization. A Cauchy combination of CUSUM statistics across multiple ridge parameters is employed, and calibration is achieved directly—without bootstrap—by leveraging a weighted bridge limit distribution. For multiple change points, a wild binary segmentation algorithm coupled with local refinement is implemented. Key contributions include the first integration of Cauchy combination and weighted bridge limits into functional change-point testing, a data-driven ridge parameter selection mechanism, and theoretical guarantees on test validity, local power, and consistent estimation of both the number and locations of change points under mild conditions. Simulations and real-data analyses demonstrate excellent finite-sample performance.

Abstract

We propose a unified ridge-regularized Hotelling framework for detecting and locating mean changes in functional time series. A growing basis expansion converts the functional observations into high-dimensional score vectors. Their long-run covariance is estimated by an edge-corrected difference-based procedure. Ridge regularization stabilizes inference under spectral decay. An explicit local-power formula shows that the power-maximizing ridge depends on the unknown spectral orientation of the change. We therefore combine a family of ridge CUSUM statistics by a Cauchy transform and calibrate the aggregate directly from their joint weighted-bridge limit. For multiple changes, we embed local maximum-ridge statistics in a wild binary segmentation procedure, followed by local refinement. Under mild conditions, we establish the validity, local power, consistency, and localization properties of the proposed tests. In the multiple-change setting, the procedure consistently recovers the number of changes and uniformly estimates their locations. The framework accommodates weak dependence and non-Gaussian functional errors. Simulations and two empirical applications demonstrate the favorable finite-sample performance of the proposed methods.

Problem

• change-point detection
• functional data
• mean shift
• time series
• high-dimensional inference

Innovation

• ridge regularization
• functional change-point detection
• Hotelling's T-squared
• wild binary segmentation
• long-run covariance estimation