Optimal covariance change point localization in high dimensions
Statistical change point analysis is concerned with identifying abrupt changes in the data that are due to actual changes in the underlying distribution. One of the main goals in the analysis is the estimation of the localization (i.e. positions of the change points). The paper under review studies the problem of change point localization in a time series of length \(n\) of independent \(p\)-dimensional random vectors with covariance matrices that are piecewise constant, and only change at unknown times. Here all the parameters quantifying the difficulty of the problem (namely the dimension \(p\), the minimal spacing, the minimal jump size, and the sub-Gaussian variance factor) change with the sample size \(n\). The authors review related literature and then propose and analyze two algorithms for covariance change point localization. Under suitable conditions it is proved that both algorithms can consistently estimate the change points. As to localization rate, the first algorithm is sub-optimal (that exhibits an unfavorable dependence on the dimension \(p\)), while the second algorithm (under a set of different and milder assumptions) yields almost minimax rate-optima. As a phase transition effect over the space of the model parameters demonstrates, ``it delivers optimal performance over nearly all scalings, for which consistent localization is possible''.
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