Gaussian and bootstrap approximations for high-dimensional U-statistics and their applications

Abstract

This paper studies the Gaussian and bootstrap approximations for the probabilities of a non-degenerate U-statistic belonging to the hyperrectangles in ${\mathbb{R}}^d$ when the dimension $d$ is large. A two-step Gaussian approximation procedure that does not impose structural assumptions on the data distribution is proposed. Subject to mild moment conditions on the kernel, we establish the explicit rate of convergence uniformly in the class of all hyperrectangles in ${\mathbb{R}}^d$ that decays polynomially in sample size for a high-dimensional scaling limit, where the dimension can be much larger than the sample size. We also provide computable approximation methods for the quantiles of the maxima of centered U-statistics. Specifically, we provide a unified perspective for the empirical bootstrap, the randomly reweighted bootstrap, and the Gaussian multiplier bootstrap with the jackknife estimator of covariance matrix as randomly reweighted quadratic forms and we establish their validity. We show that all three methods are inferentially first-order equivalent for high-dimensional U-statistics in the sense that they achieve the same uniform rate of convergence over all $d$-dimensional hyperrectangles. In particular, they are asymptotically valid when the dimension $d$ can be as large as $O(e^{n^c})$ for some constant $c\in (0,1/7)$. (Full abstract can be found in the paper.)

This paper is accepted for publication in the Annals of Statistics and it supersedes the arXiv preprint "Gaussian approximation for the sup-norm of high-dimensional matrix-variate U-statistics and its applications" (arXiv:1602.00199)

References in corpus (10)

• Regularized estimation of large covariance matrices
• Covariance regularization by thresholding
• Sparse permutation invariant covariance estimation
• Operator norm consistent estimation of large-dimensional sparse covariance matrices
• Concentration around the mean for maxima of empirical processes
• Optimal rates of convergence for sparse covariance matrix estimation
• Factor modeling for high-dimensional time series: Inference for the number of factors
• On weighted U-statistics for stationary processes
• Bootstrapping High Dimensional Time Series
• Gaussian approximation for the sup-norm of high-dimensional matrix-variate U-statistics and its applications

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