Title: Tail Bounds for Canonical $U$-Statistics and $U$-Processes with Unbounded Kernels Show Chinese title

Title: 尾界对于典范的$U$-统计量和核函数无界的$U$-过程

Abstract: In this paper, we prove exponential tail bounds for canonical (or degenerate) $U$-statistics and $U$-processes under exponential-type tail assumptions on the kernels. Most of the existing results in the relevant literature often assume bounded kernels or obtain sub-optimal tail behavior under unbounded kernels. We obtain sharp rates and optimal tail behavior under sub-Weibull kernel functions. Some examples from nonparametric and semiparametric statistics literature are considered. Show Chinese abstract

Abstract: 本文中，我们证明了在核函数满足指数型尾部假设下，典范（或退化）$U$-统计量和$U$-过程的指数尾界。大多数现有文献中的结果通常假设核函数有界，或者在核函数无界时得到次最优的尾部行为。我们在次韦伯尔核函数下获得了尖锐的速率和最优的尾部行为。考虑了一些来自非参数和半参数统计学文献的例子。