Abstract:
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We studied two problems for high-dimensional change point inference and identification in mean vectors based on the supremum norm of the CUSUM statistics with a boundary removal parameter. For the problem of testing for existence of a mean-shift in a sequence of independent observations, we introduce a Gaussian multiplier bootstrap to approximate the critical values of the CUSUM test statistics when dimension $p$ is much larger than $n$. Under arbitrary dependence structures and mild moment conditions, our test enjoys the uniform validity in size under the null and is powerful under the sparse alternatives. Once a change point is detected, the location is estimated by maximizing the sup-norm of generalized CUSUM statistics at two different weighting scales. In both of the estimators that relied on the covariance stationary CUSUM statistics and the non-stationary one with less weights on boundary points, consistency is achieved in possible optimal rate of $n$ for sub-exponential case up to a log factor, only through which $p$ impacts the convergence rate. The results are non-asymptotic, data-dependent and simulations in finite samples show an encouraging agreement with theoretics.
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