Tail Bounds for Canonical $U$-Statistics and $U$-Processes with Unbounded Kernels

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.

This is a slightly edited version of the 2018 draft available at https://faculty.wharton.upenn.edu/wp-content/uploads/2018/10/Chakrabortty-UStat-Draft.pdf. Added more comments on the assumptions and the proof technique of Theorem 1. Corrected a few typos. More improvements to follow in the future for the U-process results