Inference on the Change Point in a High Dimensional Covariance Shift
Authors: Abhishek Kaul, Hongjin Zhang, Konstantinos Tsampourakis, George Michailidis; Journal of Machine Learning Research, 24(168):1−68, 2023.
Abstract
This study addresses the problem of constructing confidence intervals for the change point in a high-dimensional covariance shift scenario. A new estimator for the change point parameter is introduced, and its asymptotic distribution is derived under high dimensional scaling. The proposed estimator demonstrates a rapid rate of convergence at $O_p(\psi^{-2})$, where $\psi$ represents the jump size between model parameters before and after the change point. The asymptotic distributions are characterized for both vanishing and non-vanishing regimes of the jump size. In the former case, the distribution corresponds to the argmax of an asymmetric Brownian motion, while in the latter case, it corresponds to the argmax of an asymmetric random walk. Furthermore, the relationship between these distributions is established, enabling the construction of regime-adaptive confidence intervals. Simple and effective algorithms for implementing the proposed methodology are developed, and their performance is demonstrated on synthetic and real datasets.
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