Finite sample change point inference and identification for high-dimensional mean vectors
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Cumulative sum (CUSUM) statistics are widely used in the change point inference and identification. This paper studies the two problems for high-dimensional mean vectors based on the supremum norm of the CUSUM statistics. For the problem of testing for the existence of a change point in a sequence of independent observations generated from the mean-shift model, we introduce a Gaussian multiplier bootstrap to approximate critical values of the CUSUM test statistics in high dimensions. The proposed bootstrap CUSUM test is fully data-dependent and it has strong theoretical guarantees under arbitrary dependence structures and mild moment conditions. Specifically, we show that with a boundary removal parameter the bootstrap CUSUM test enjoys the uniform validity in size under the null and it achieves the minimax separation rate under the sparse alternatives when the dimension p can be larger than the sample size n. Once a change point is detected, we estimate the change point location by maximizing the supremum norm of the generalized CUSUM statistics at two different weighting scales. The first estimator is based on the covariance stationary CUSUM statistics at each data point, which is consistent in estimating the location at the nearly parametric rate n^-1/2 for sub-exponential observations. The second estimator is a non-stationary CUSUM statistics, assigning less weights on the boundary data points. In the latter case, we show that it achieves the nearly best possible rate of convergence on the order n^-1. In both cases, the dimension impacts the rate of convergence only through the logarithm factors, and therefore consistency of the CUSUM location estimators is possible when p is much larger than n.
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