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Self-Normalization for CUSUM-based Change Detection in Locally Stationary Time Series

Florian Heinrichs

TL;DR

A new self-normalized CUSUM test is proposed for detecting changes in the mean of a locally stationary time series, based on a bivariate partial-sum process, and it is shown that the resulting self-normalized test attains asymptotic level $\alpha$ under the null hypothesis of no change.

Abstract

A new self-normalized CUSUM test is proposed for detecting changes in the mean of a locally stationary time series. For stationary data, self-normalization relies on the factorization of a constant long-run variance and a stochastic factor. In this case, the CUSUM statistic can be divided by another statistic proportional to the long-run variance, so that the latter cancels, avoiding estimation of the long-run variance. Under local stationarity, the partial sum process converges to $\int_0^t σ(x) d B_x$ and no such factorization is possible. To overcome this obstacle, a self-normalized test statistic is introduced, based on a bivariate partial-sum process. Weak convergence of the process is proven, and it is shown that the resulting self-normalized test attains asymptotic level $α$ under the null hypothesis of no change, while being consistent against abrupt, gradual, and multiple changes under mild assumptions. Simulation studies show that the proposed test has accurate size and substantially improved finite-sample power relative to existing approaches. Two data examples illustrate practical performance.

Self-Normalization for CUSUM-based Change Detection in Locally Stationary Time Series

TL;DR

A new self-normalized CUSUM test is proposed for detecting changes in the mean of a locally stationary time series, based on a bivariate partial-sum process, and it is shown that the resulting self-normalized test attains asymptotic level under the null hypothesis of no change.

Abstract

A new self-normalized CUSUM test is proposed for detecting changes in the mean of a locally stationary time series. For stationary data, self-normalization relies on the factorization of a constant long-run variance and a stochastic factor. In this case, the CUSUM statistic can be divided by another statistic proportional to the long-run variance, so that the latter cancels, avoiding estimation of the long-run variance. Under local stationarity, the partial sum process converges to and no such factorization is possible. To overcome this obstacle, a self-normalized test statistic is introduced, based on a bivariate partial-sum process. Weak convergence of the process is proven, and it is shown that the resulting self-normalized test attains asymptotic level under the null hypothesis of no change, while being consistent against abrupt, gradual, and multiple changes under mild assumptions. Simulation studies show that the proposed test has accurate size and substantially improved finite-sample power relative to existing approaches. Two data examples illustrate practical performance.

Paper Structure

This paper contains 16 sections, 11 theorems, 181 equations, 5 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Bivariate Partial Sum Process
  3. Detecting Change Points and Gradual Changes
  4. Local Alternatives and Monotone Power
  5. Estimating the First Point of Change
  6. Empirical Results
  7. Simulation Study
  8. Local Alternatives
  9. Case Study
  10. Conclusion
  11. Proofs
  12. Auxiliary Results
  13. Proof of Lemma \ref{['lem:fdd_conv']}
  14. Proof of Lemma \ref{['lem:equicont']}
  15. Proof of Results from Section \ref{['sec:change_detection']}
  16. ...and 1 more sections

Key Result

Theorem 5

Let Assumptions assump:error and assump:sequences be satisfied. Then, the centered partial sum process $G_n = \{G_n(t,s)\}_{t, s\in[0, 1]}$, with converges weakly to $\{G(t, s)\}_{t, s\in[0, 1]}$, where for a standard Brownian sheet $B$.

Figures (5)

  • Figure 1: Visualization of indices of the bivariate partial sum process
  • Figure 2: Various mean functions, used to generate time series under the alternative.
  • Figure 3: Exemplary trajectories of $AR(1)$ processes with $\sigma= 2\sigma_2$ (left) and $\sigma= 2\sigma_3$ (right), for $\mu(x) \equiv 0$ and $n=200$.
  • Figure 4: Empiricial rejection rates ($y$-axis) across different values of $a_n$ (logarithmic $x$-axis) for local alternatives. The horizontal line at $5\%$ indicates the nominal level under the null hypothesis.
  • Figure 5: Mean temperatures in Gayndah (left), Robe (center) and Sydney (right) for the month of July (gray), jointly with overall averages (black).

Theorems & Definitions (12)

  • Remark 4
  • Theorem 5
  • Lemma 6
  • Lemma 7
  • Corollary 8
  • Corollary 9
  • Corollary 10
  • Corollary 11
  • Proposition 12
  • Proposition 13
  • ...and 2 more