Gaussian Approximation for High-Dimensional Second-Order $U$- and $V$-statistics with Size-Dependent Kernels under i.n.i.d. Sampling

TL;DR

This paper develops Gaussian approximation methods for high-dimensional second-order $U$- and $V$-statistics with size-dependent kernels under i.n.i.d. sampling, covering a broad range of statistical models and tests.

Contribution

It introduces Gaussian approximation techniques for high-dimensional $U$- and $V$-statistics with size-dependent kernels in the i.n.i.d. setting, extending existing methods.

Findings

Valid Gaussian approximations for high-dimensional vectors of $U$- and $V$-statistics.

Extension of maximal inequalities to the i.n.i.d. setting.

Applicability to various regression and two-sample testing scenarios.

Abstract

We develop Gaussian approximations for high-dimensional vectors formed by second-order $U$- and $V$-statistics whose kernels depend on sample size under independent but not identically distributed (i.n.i.d.) sampling. Our results hold irrespective of which component of the Hoeffding decomposition is dominant, thereby covering both non-degenerate and degenerate regimes as special cases. By allowing i.n.i.d.~sampling, the class of statistics we analyze includes weighted $U$- and $V$-statistics and two-sample $U$- and $V$-statistics as special cases, which cover estimators of parameters in regression models with many covariates, many-weak instruments as well as a broad class of smoothed two-sample tests and the separately exchangeable arrays, among others. In addition, we extend sharp maximal inequalities for high-dimensional $U$-statistics with size-dependent kernels from the i.i.d.~to the i.n.i.d.~setting, which may be of independent interest.