标题： 高维$U$-统计量具有大小相关核的高斯近似 显示英文标题

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

摘要： 受计量经济学中基于核函数的半参数估计量在小带宽渐近性方面的启发，本文建立了关于高维固定阶数的$U$-统计量的Gaussian逼近结果，这些统计量的核函数依赖于样本大小。 我们的结果允许Hoeffding分解的主要成分不存在或未知的情况，包括一些已知退化程度的特殊情况。 对于Gaussian逼近所得到的误差界足够尖锐，当应用于一个典型的半参数估计问题时，在固定维度设置下几乎可以恢复小带宽渐近性的最弱带宽条件。 我们还展示了适应性拟合优度检验的应用，并讨论了几个潜在的应用。 显示英文摘要

摘要： 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, along with discussions about several potential applications.