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

Shunsuke Imai

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.

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

Abstract

We develop Gaussian approximations for high-dimensional vectors formed by second-order - and -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 - and -statistics and two-sample - and -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 -statistics with size-dependent kernels from the i.i.d.~to the i.n.i.d.~setting, which may be of independent interest.

Paper Structure

This paper contains 48 sections, 22 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. Organization:
  3. Notations:
  4. Notation and Setting
  5. Notations for Second-Order $U$-statistics:
  6. Main Results
  7. Gaussian Approximation for High-dimensional $U$-statistics
  8. Gaussian Approximation for High-dimensional $V$-statistics
  9. Maximal Inequalities
  10. Special Cases and Potential Applications
  11. Weighted $U$-statistics
  12. Many-Weak Instrumental Variables Asymptotics in MAM framework
  13. Literature :
  14. Framework :
  15. Many Covariates Asymptotics in MAM framework
  16. ...and 33 more sections

Key Result

Theorem 1

Assume $\max_{j\in[p]}\max_{(i,m)\in I_{n,2}}\|\psi_{j,(i,m)}\|_{L^q(P_i\otimes P_m)} < \infty$ for some $q\in[4,\infty]$. Then, there exists a universal constant $C$ such that where $Z\sim N(0, \mathop{\mathrm{Cov}}\nolimits(W))$ and

Theorems & Definitions (58)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • proof
  • Corollary 3
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 1
  • Remark 2: Weighted $U$-statistics as $U$-statistics under i.n.i.d. sampling
  • ...and 48 more