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

By Xiaohui Chen. 2016

64 citations recorded by the source corpus.

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.)

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c47dc03229814f2355da09ffcdf402c058f23fd9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Statistics and Probability

Citations in the corpus

High-dimensional econometrics and regularized GMM (2018, methodology): “[42, 43] develop extensions of the high-dimensional”
Finite sample change point inference and identification for high‐dimensional mean vectors (2017, background): “Bootstrap approximations have been extended to partial sums of weakly dependent time series in [50, 52] and more general nonlinear statistics in [14].”
High-dimensional central limit theorems by Stein’s method (2020, background): “…and Kato (2017a), Chernozhukov et al. (2019) for generalization to normal approximation on hyperrectangles and improvements of the error bound, Chen (2018), Chen and Kato (2019), Song, Chen and Kato (2019) for extensions to U -statistics, Chernozhukov, Chetverikov and Kato (2019), Zhang and…”