Cram\'er type moderate deviation theorems for self-normalized processes

TL;DR

This paper develops new theoretical results, including a concentration inequality and a Cramér type moderate deviation theorem, for self-normalized processes and Studentized U-statistics, enhancing the understanding of their normal approximation accuracy.

Contribution

It introduces a novel randomized concentration inequality and establishes a sharp moderate deviation theorem for self-normalized processes under optimal moment conditions.

Findings

Established a Cramér type moderate deviation theorem for self-normalized processes.

Developed a new randomized concentration inequality.

Provided sharp results for Studentized U-statistics.

Abstract

Cram\'er type moderate deviation theorems quantify the accuracy of the relative error of the normal approximation and provide theoretical justifications for many commonly used methods in statistics. In this paper, we develop a new randomized concentration inequality and establish a Cram\'er type moderate deviation theorem for general self-normalized processes which include many well-known Studentized nonlinear statistics. In particular, a sharp moderate deviation theorem under optimal moment conditions is established for Studentized $U$-statistics.