🤖 AI Summary
Conventional self-normalized CUSUM methods fail for mean change-point detection in locally stationary time series because they cannot disentangle long-run variance from stochastic fluctuations. Method: We propose a novel bivariate partial-sum-process-driven self-normalized CUSUM procedure that jointly constructs a mean-shift component and a variance-correction component, enabling asymptotically exact inference without estimating the long-run variance. Contribution/Results: Under local stationarity, the method achieves asymptotic α-level control and consistency. Its limiting distribution is derived via weak convergence theory and approximated by the integral of Brownian motion. Monte Carlo simulations demonstrate substantial finite-sample improvements over existing approaches. Empirical analyses on real financial and climate datasets confirm its robustness in detecting mean shifts.
📝 Abstract
A novel self-normalization procedure for CUSUM-based change detection in the mean of a locally stationary time series is introduced. Classical 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. Thereby, a tedious estimation of the long-run variance can be avoided. 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 constructed from a carefully designed 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 a broad class of alternatives. Extensive simulations demonstrate better finite-sample properties compared to existing methods. Applications to real data illustrate the method's practical effectiveness.