lp-Statistics in High Dimensions

This paper considers a new bootstrap procedure to estimate the distribution of highdimensional lp-statistics, i.e. the lp-norms of the sum of n independent d-dimensional random vectors with d ≫ n and p ∈ [1,∞]. We provide a non-asymptotic characterization of the sampling distribution of lp-statistics based on Gaussian approximation and show that the bootstrap procedure is consistent in the Kolmogorov-Smirnov distance under mild conditions on the covariance structure of the data. As an application of the general theory we propose a bootstrap hypothesis test for simultaneous inference on high-dimensional mean vectors. We establish its asymptotic correctness and consistency under high-dimensional alternatives, and discuss the power of the test as well as the size of associated confidence sets. We illustrate the bootstrap and testing procedure numerically on simulated data.

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