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

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.