Xiaohui Chen
University of Illinois at Urbana-Champaign

Abstract: Recent years have seen tremendous progress in quantifying the uncertainty of statistical inference for high-dimensional (large d) and big (large n) data. In this talk, we will first discuss non-asymptotic Gaussian and bootstrap approximations of U-statistics on high-dimensional hyperrectangles. U-statistics form a rich class of nonlinear statistics that play a key role in many important problems including estimating covariance and rank correlation matrices, testing monotonicity or concavity of nonparametric regression functions, robust change point detection, random forests, among others. Our Gaussian approximation bounds do not make structural assumptions (such as sparsity) and are dimension-free, thus enabling us to handle a broader spectrum of applications without model misspecification and allowing for infinite-dimensional generalizations to principled nonparametric inference. On the other hand, data-dependent inferential procedures for U-statistics are often computationally prohibitive. To overcome such computational bottlenecks, we introduce randomized incomplete U-statistics whose computational cost can be made independent of its order with guaranteed statistical validity.