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.

翻译：本文提出了一种新的自标准化CUSUM检验方法，用于检测局部平稳时间序列均值的变化。对于平稳数据，自标准化依赖于常数长期方差与随机因子的分解。在这种情况下，CUSUM统计量可以除以另一个与长期方差成比例的统计量，从而消除后者，避免了长期方差的估计。在局部平稳条件下，部分和过程收敛于$\int_0^t σ(x) d B_x$，无法进行此类分解。为克服这一障碍，本文基于二元部分和过程引入了一种自标准化检验统计量。证明了该过程的弱收敛性，并表明所得的自标准化检验在无变化的原假设下达到渐近水平$α$，同时在温和假设下对突变、渐变及多重变化具有一致性。模拟研究表明，相较于现有方法，所提检验具有精确的尺寸和显著提升的有限样本功效。两个数据实例展示了其实际性能。