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.

翻译：受计量经济学中基于核的半参数估计量的小带宽渐近理论启发，本文建立了针对高维固定阶$U$-统计量的高斯逼近结果，其核函数依赖于样本量。我们的结果允许存在Hoeffding分解中主导成分缺失或未知的情形，并将已知退化度的情形作为特例包含在内。所获得的高斯逼近误差界足够精确，当应用于典型半参数估计问题时，几乎能恢复固定维情形下小带宽渐近理论中最弱的带宽条件。我们还提出了该方法在自适应拟合优度检验以及高维密度加权平均导数同时推断中的应用，并讨论了若干潜在的应用方向。