While there is considerable work on change point analysis in univariate time series, more and more data being collected comes from high dimensional multivariate settings. This paper introduces the asymptotic concept of high dimensional efficiency which quantifies the detection power of different statistics in such situations. While being related to classic asymptotic relative efficiency, it is different in that it provides the rate at which the change can get smaller with dimension while still being detectable. This also allows for comparisons of different methods with different null asymptotics as is for example the case in high-dimensional change point settings. Based on this new concept we investigate change point detection procedures using projections and develop asymptotic theory for how full panel (multivariate) tests compare with both oracle and random projections. Furthermore, for each given projection we can quantify a cone such that the corresponding projection statistic yields better power behavior if the true change direction is within this cone. The effect of misspecification of the covariance on the power of the tests is investigated, because in many high dimensional situations estimation of the full dependency (covariance) between the multivariate observations in the panel is often either computationally or even theoretically infeasible. It turns out that the projection statistic is much more robust in this respect in terms of size and somewhat more robust in terms of power. The theoretic quantification by the theory is accompanied by simulation results which confirm the theoretic (asymptotic) findings for surprisingly small samples. This shows in particular that the concept of high dimensional efficiency is indeed suitable to describe small sample power, and this is demonstrated in a multivariate example of market index data.
Rank-based change-point analysis for long-range dependent time series
我们考虑基于秩统计量的变点检验来检验长期相关观测值的结构变化。在平稳时间序列的假设下，在变点高度随变点高度减小而变化的假设下，导出了相应的检验统计量的渐近分布。为此，证明了序列经验过程在具有加权上确界范数的两参数Skorohod空间中的一致约化原理。此外，我们比较了秩检验的效率所产生的考虑不同的评分函数。在高斯性下，基于秩的检验相对于CuSum检验的渐近相对效率为1，与评分函数无关。在秩基变点检验的实际应用中，我们建议将自归一化秩统计量与二次抽样相结合。理论结果是伴随着模拟研究，特别是，允许比较秩检验从不同的评分函数。对于基于秩的变点检验的有限样本性能，Van der Waerden秩检验被证明在广泛的情况下是有利的。最后，我们分析了经济，水文和网络流量监测的数据集的结构变化，并将我们的结果与以前的数据分析进行比较。
