Online Monitoring of Structural Change Points Based on Ratio-Type Statistics

For scenarios where the type of structural break in a time series is unknown, this paper proposes a modified ratio-type test statistic to enable effective online monitoring of structural breaks, while circumventing the estimation of long-term variance. Under specific assumptions, we rigorously derive the asymptotic distribution of the test statistic under the null hypothesis and establish its consistency under the alternative hypothesis. In cases where both variance and mean breaks coexist, we introduce a refined mixed-break monitoring procedure based on the consistent estimation of breakpoints. The proposed method first provides consistent estimations of the mean change points and variance change points separately; then, mean and variance removal are performed on original data; finally, the previously removed trend is added back. Compared to traditional monitoring methods, which have to use two test statistics, this method requires only one to simultaneously monitor both types of change points, resulting in a significantly simplified monitoring process. This approach effectively reduces mutual interference between the two types of breaks, thereby enhancing the power of the test. Extensive numerical simulations confirm that this method can accurately detect the presence of structural breaks and reliably identify their types. Finally, case studies are provided to demonstrate the efficacy and practical applicability of the proposed method.