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Multiple change-points detection based on U-Statistics under weak dependence

Joseph Ngatchou-Wandji, Echarif Elharfaoui, Michel Harel

TL;DR

This work advances change-point analysis by developing a multichange-point framework based on $U$-statistics for weakly dependent data. By formulating a process $Z_n(t_1, t_k)$ and KS/CV-type statistics, the paper derives the limiting behavior under both the null and local alternatives, with explicit Gaussian and drift components, and provides practical methods for critical-value calibration via simulation. The results cover general kernels and special anti-symmetric cases, yielding distribution-free limits in the latter and enabling efficient testing for multiple mean, variance, and autocorrelation changes. The simulations illustrate competitive or superior performance against classical methods, while also highlighting the impact of kernel choice on sensitivity to different types of changes. Overall, the paper offers a rigorous, extensible toolkit for detecting multiple, possibly heterogeneous change-points under weak dependence.

Abstract

We study multiple change-points detection using multi-samples tests based on U-statistics for absolutely regular observations. Our results extend those of Ngatchou-Wandji et al. (2022) concerned with the study of one single changepoint. The asymptotic distributions of the test statistics under the null hypothesis and under a sequence of local alternatives are given explicitly, and the tests are shown to be consistent. A small set of simulations is done for evaluating the performance of the tests in detecting multiple changes in the mean, variance and autocorrelation of some simple times series models.

Multiple change-points detection based on U-Statistics under weak dependence

TL;DR

This work advances change-point analysis by developing a multichange-point framework based on -statistics for weakly dependent data. By formulating a process and KS/CV-type statistics, the paper derives the limiting behavior under both the null and local alternatives, with explicit Gaussian and drift components, and provides practical methods for critical-value calibration via simulation. The results cover general kernels and special anti-symmetric cases, yielding distribution-free limits in the latter and enabling efficient testing for multiple mean, variance, and autocorrelation changes. The simulations illustrate competitive or superior performance against classical methods, while also highlighting the impact of kernel choice on sensitivity to different types of changes. Overall, the paper offers a rigorous, extensible toolkit for detecting multiple, possibly heterogeneous change-points under weak dependence.

Abstract

We study multiple change-points detection using multi-samples tests based on U-statistics for absolutely regular observations. Our results extend those of Ngatchou-Wandji et al. (2022) concerned with the study of one single changepoint. The asymptotic distributions of the test statistics under the null hypothesis and under a sequence of local alternatives are given explicitly, and the tests are shown to be consistent. A small set of simulations is done for evaluating the performance of the tests in detecting multiple changes in the mean, variance and autocorrelation of some simple times series models.

Paper Structure

This paper contains 13 sections, 15 theorems, 120 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Definitions and assumptions
  3. Asymptotics
  4. Results for general kernels
  5. Results for some particular kernels
  6. Simulation study
  7. Concluding remarks
  8. Appendix : Proofs of the results
  9. Some useful results
  10. Preliminaries
  11. Proof of Theorem \ref{['th1']}
  12. Proof of Theorem \ref{['th2']}
  13. Proof of Theorem \ref{['th3']}

Key Result

Theorem 1

Assume that (A1)-(A2) hold. Then, under $\mathcal{H}_{0}$, if for some $\delta >0$ then for any $p,r= 1, 2$, $\sigma_{pr}<\infty$. If in addition $\sigma_{pr}>0$, $1\leq p,r\leq2$, then $\{\mathcal{Z}_{n}(t_1,t_2,\ldots,t_k); t_1,t_2, \ldots, t_k \in [0,1]\}_{n\in \mathbb{N}}$ converges in distribution towards the process $\mathcal{Z}(\cdot)$ defined for any $t=(t_1, \ldots, t_k) where by convent

Figures (1)

  • Figure 1: Chronograms of series with two changes. On the first row, the first graphic is that of a series with two changes in the mean. The second is that of a series with two changes in the autocorrolation, the last is that of a series with two changes in the variance. On the second row, the first chronogram is that of a series with one change in the mean and one in the autocorrelation. The next is that of a series with one changes in the mean and one in the variance, and the last is that of a series with one change in the variance and another one in the autocorrelation.

Theorems & Definitions (21)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Theorem 7
  • ...and 11 more