Cauchy Aggregation of Ridge-Regularized Hotelling Tests for High-Dimensional Change-Point Detection

2026-06-12

AI Summary

In high-dimensional change-point detection, the optimal ridge parameter for ridge-regularized Hotelling’s test depends on the unknown covariance structure and mean shift, making it difficult to specify a priori. This work proposes a novel approach that circumvents the need to select a single ridge parameter by computing p-values across a deterministic grid of fixed ridge parameters and aggregating them via the Cauchy combination rule—a technique introduced here for the first time in this context. Leveraging random matrix theory, the method establishes a joint weak convergence analysis that guarantees valid type-I error control while achieving power nearly equivalent to that of the optimal fixed ridge choice. Monte Carlo experiments demonstrate its robust performance across diverse covariance structures and signal configurations, highlighting both theoretical rigor and practical effectiveness.

Abstract

Ridge-regularized Hotelling-type (RHT) change-point tests depend on a ridge parameter $λ$, but the power-optimal value is determined by the unknown covariance structure and the unknown mean shift. We avoid selecting a single ridge value by computing fixed-ridge p-values on a finite deterministic grid and aggregating them with the Cauchy combination rule. Under the standard random-matrix conditions for fixed-ridge RHT statistics, we establish finite-grid joint weak convergence of the ridge processes. This leads to fixed-level validity under joint-limit calibration and small-tail validity for the analytic Cauchy p-value. Monte Carlo experiments show that deterministic-grid Cauchy aggregation has stable size behavior and achieves power close to the best stable fixed ridge choice across a range of covariance and signal configurations.

Problem

• change-point detection
• ridge regularization
• Hotelling test
• high-dimensional data
• covariance structure

Innovation

• Cauchy aggregation
• ridge regularization
• Hotelling test
• change-point detection
• high-dimensional statistics