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Gaussian Approximation for High-Dimensional $U$-statistics with Size-Dependent Kernels

Shunsuke Imai, Yuta Koike

TL;DR

This paper develops Gaussian-approximation results for high-dimensional fixed-order $U$-statistics with kernels that may depend on the sample size, accommodating cases where the dominant Hoeffding projection is unknown or degenerate. It introduces a generalized exchangeable-pairs approach and sharp maximal inequalities to obtain explicit nonasymptotic error bounds, with refinements for the $r=2$ case and moment-based corollaries. The theory is illustrated through a toy example on average marginal densities and applied to adaptive goodness-of-fit testing and high-dimensional inference for density-weighted averaged derivatives (DWADs), including an empirical DWAD analysis of price elasticities in India. The results enable adaptive, simultaneous inference in settings where the dimension grows with the sample size, without requiring asymptotic linearity, and they recover near-optimal bandwidth conditions up to logarithmic factors. Overall, the framework broadens the scope of reliable inference for complex, high-dimensional semiparametric estimators and related two-step procedures.

Abstract

Motivated by small bandwidth asymptotics for kernel-based semiparametric estimators in econometrics, this paper establishes Gaussian approximation results for high-dimensional fixed-order $U$-statistics whose kernels depend on the sample size. Our results allow for a situation where the dominant component of the Hoeffding decomposition is absent or unknown, including cases with known degrees of degeneracy as special forms. The obtained error bounds for Gaussian approximations are sharp enough to almost recover the weakest bandwidth condition of small bandwidth asymptotics in the fixed-dimensional setting when applied to a canonical semiparametric estimation problem. We also present an application to an adaptive goodness-of-fit testing and the simultaneous inference on high-dimensional density weighted averaged derivatives, along with discussions about several potential applications.

Gaussian Approximation for High-Dimensional $U$-statistics with Size-Dependent Kernels

TL;DR

This paper develops Gaussian-approximation results for high-dimensional fixed-order -statistics with kernels that may depend on the sample size, accommodating cases where the dominant Hoeffding projection is unknown or degenerate. It introduces a generalized exchangeable-pairs approach and sharp maximal inequalities to obtain explicit nonasymptotic error bounds, with refinements for the case and moment-based corollaries. The theory is illustrated through a toy example on average marginal densities and applied to adaptive goodness-of-fit testing and high-dimensional inference for density-weighted averaged derivatives (DWADs), including an empirical DWAD analysis of price elasticities in India. The results enable adaptive, simultaneous inference in settings where the dimension grows with the sample size, without requiring asymptotic linearity, and they recover near-optimal bandwidth conditions up to logarithmic factors. Overall, the framework broadens the scope of reliable inference for complex, high-dimensional semiparametric estimators and related two-step procedures.

Abstract

Motivated by small bandwidth asymptotics for kernel-based semiparametric estimators in econometrics, this paper establishes Gaussian approximation results for high-dimensional fixed-order -statistics whose kernels depend on the sample size. Our results allow for a situation where the dominant component of the Hoeffding decomposition is absent or unknown, including cases with known degrees of degeneracy as special forms. The obtained error bounds for Gaussian approximations are sharp enough to almost recover the weakest bandwidth condition of small bandwidth asymptotics in the fixed-dimensional setting when applied to a canonical semiparametric estimation problem. We also present an application to an adaptive goodness-of-fit testing and the simultaneous inference on high-dimensional density weighted averaged derivatives, along with discussions about several potential applications.

Paper Structure

This paper contains 59 sections, 32 theorems, 422 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Literature and Technical Contributions:
  3. Organization:
  4. General notation and convention:
  5. Main results
  6. $U$-statistics related notation
  7. Main results
  8. An illustration: Estimation of the average marginal densities
  9. Application 1: Adaptive goodness-of-fit tests
  10. Application 2: Inference on many density weighted average derivatives
  11. Jackknife multiplier bootstrap for many density weighted average derivatives
  12. Empirical Application : Price Elasticities Matrices of food in India 2022
  13. Conclusion and Discussion
  14. Proofs for Section \ref{['sec:main']}
  15. High-dimensional CLTs via generalized exchangeable pairs
  16. ...and 44 more sections

Key Result

Theorem 1

There exists a constant $C_r$ depending only on $r$ such that where $Z\sim N(0,\mathop{\mathrm{Cov}}\nolimits[W])$.

Figures (1)

  • Figure 1: Heatmaps of DWAD estimates for price-elasticities

Theorems & Definitions (69)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Remark 1: Sufficient conditions for convergence of the bound
  • Remark 2: Sub-Weibull case
  • Corollary 2
  • Remark 3: Sufficient conditions for convergence of the bound
  • Remark 4: Relation to cattaneo2014small
  • Remark 5: Application of \ref{['coro:clt-ustat']}
  • Proposition 1
  • ...and 59 more