Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sequential Eigenvalue Statistics for Change-Point Detection in Covariance Matrices

Nina Dörnemann, Holger Dette

TL;DR

This work tackles change-point detection in covariance structures within a moderately high-dimensional regime where $p/n \to y$. It introduces a sequential min-type likelihood-ratio statistic built from centered, sequential LRTs and proves a null limit in which the test statistic converges to a nuisance-free Gaussian process $Z(t)$, enabling straightforward critical values. A novel, consistent estimator of kurtosis $\hat{\kappa}_n$ is developed to stabilize centering and variance in the asymptotics. Finite-sample simulations show accurate size control and superior power relative to existing high-dimensional tests, and a dedicated change-point estimator $\hat{\tau}^*$ achieves lower bias and MSE than competing methods. The approach does not rely on sparsity or sub-Gaussian assumptions and leverages random-matrix theory to capture the dependency structure of sequential statistics, offering practical tools for covariance-change detection in moderately high dimensions.

Abstract

Testing for change points in sequences of covariance matrices is an important and equally challenging problem in statistical methodology with applications in various fields. Motivated by the observation that even in cases where the ratio between dimension and sample size is as small as $0.05$, tests based on a fixed-dimension asymptotics do not keep their preassigned level, we propose to derive critical values of test statistics using an asymptotic regime where the dimension diverges at the same rate as the sample size. This paper introduces a novel and well-founded statistical methodology for detecting change points in a sequence of moderately dimensional covariance matrices. Our approach utilizes a min-type statistic based on a sequential process of likelihood ratio statistics. This is used to construct a test for the hypothesis of the existence of a change point with a corresponding estimator for its location. We provide theoretical guarantees by thoroughly analyzing the asymptotic properties of the sequential process of likelihood ratio statistics. In particular, we prove weak convergence towards a Gaussian process under the null hypothesis of no change. To identify the challenging dependency structure between consecutive test statistics, we employ tools from random matrix theory and stochastic processes.

Sequential Eigenvalue Statistics for Change-Point Detection in Covariance Matrices

TL;DR

This work tackles change-point detection in covariance structures within a moderately high-dimensional regime where . It introduces a sequential min-type likelihood-ratio statistic built from centered, sequential LRTs and proves a null limit in which the test statistic converges to a nuisance-free Gaussian process , enabling straightforward critical values. A novel, consistent estimator of kurtosis is developed to stabilize centering and variance in the asymptotics. Finite-sample simulations show accurate size control and superior power relative to existing high-dimensional tests, and a dedicated change-point estimator achieves lower bias and MSE than competing methods. The approach does not rely on sparsity or sub-Gaussian assumptions and leverages random-matrix theory to capture the dependency structure of sequential statistics, offering practical tools for covariance-change detection in moderately high dimensions.

Abstract

Testing for change points in sequences of covariance matrices is an important and equally challenging problem in statistical methodology with applications in various fields. Motivated by the observation that even in cases where the ratio between dimension and sample size is as small as , tests based on a fixed-dimension asymptotics do not keep their preassigned level, we propose to derive critical values of test statistics using an asymptotic regime where the dimension diverges at the same rate as the sample size. This paper introduces a novel and well-founded statistical methodology for detecting change points in a sequence of moderately dimensional covariance matrices. Our approach utilizes a min-type statistic based on a sequential process of likelihood ratio statistics. This is used to construct a test for the hypothesis of the existence of a change point with a corresponding estimator for its location. We provide theoretical guarantees by thoroughly analyzing the asymptotic properties of the sequential process of likelihood ratio statistics. In particular, we prove weak convergence towards a Gaussian process under the null hypothesis of no change. To identify the challenging dependency structure between consecutive test statistics, we employ tools from random matrix theory and stochastic processes.
Paper Structure (14 sections, 13 theorems, 171 equations, 1 figure, 5 tables)

This paper contains 14 sections, 13 theorems, 171 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Change point analysis by a sequential LRT process
  3. Finite-sample properties
  4. Numerical experiments for change-point detection
  5. Numerical experiments for the change-point estimation
  6. Proofs of main results under the null hypothesis
  7. Proof of Theorem \ref{['thm_main_cen']}
  8. Proof of Theorem \ref{['thm_fidis']} - weak convergence of finite-dimensional distributions
  9. Supplementary Material
  10. Auxiliary results for the proof of Theorem \ref{['thm_fidis']}
  11. Proof of Theorem \ref{['thm_tight']} - asymptotic tightness
  12. Auxiliary results
  13. Proof of Proposition \ref{['lem_consistency_kappa']}
  14. Proofs of Lemma \ref{['lem_mom_ineq_X']} - Lemma \ref{['lem_mom_ineq_quad_term_tight']}

Key Result

Proposition 1

Suppose that assumptions ass_mom-ass_sigma_null are satisfied, and $p/n\to y\in (0,\infty)$ as $n\to\infty$. Then, under $H_0$, we have

Figures (1)

  • Figure 1: Empirical rejection rates of the new test \ref{['test']} (triangle) compared to the test of ryankillick2023 (diamond), where $t_0 =0.2, ~t^\star=0.5$ and $(n,p)=(600,50)$ (first row), $(n,p)=(600,80)$ (middle row), $(n,p)=(800,100)$ (third row). Left panels: model \ref{['model1']}), where $x_{11}\sim {\cal N} (0,1)$. Right panels: model \ref{['model2']}, where $x_{11}\sim \mathcal{U}(0,1)$.

Theorems & Definitions (25)

  • Proposition 1
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Remark 2
  • proof : Proof of Theorem \ref{['thm_main_cen']}
  • proof : Proof of Theorem \ref{['thm_fidis']}
  • Lemma 1
  • ...and 15 more