Title: Finite sample change point inference and identification for high-dimensional mean vectors
Authors: Xiaohui Chen - University of Illinois at Urbana-Champaign (United States) [presenting]
Mengjia Yu - University of Illinois at Urbana-Champaign (United States)
Abstract: Cumulative sum (CUSUM) statistics are widely used in the change point inference and identification. We discuss two problems for high-dimensional mean vectors based on the max-norm of CUSUM statistics. For the problem of testing for the existence of a change point in an independent sample generated from the mean-shift model, we propose a Gaussian multiplier bootstrap to calibrate 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 max-norm of the CUSUM statistics at two different weighting scales. The first estimator is based on the covariance stationary CUSUM statistics, and the second estimator is based on non-stationary CUSUM statistics assigning less weights to the boundary data points. In the latter case, we show that it achieves the nearly best possible rate of convergence. In both cases, dimension impacts the rate of convergence only through the logarithm factors, and consistency of the CUSUM location estimators is possible when $p >> n$.