A moderate deviation principle for empirical bootstrap measure

TL;DR

This paper establishes large deviation principles for empirical bootstrap measures and their joint distributions, providing a theoretical foundation for analyzing moderate deviations in bootstrap-based statistical procedures.

Contribution

It introduces two new large deviation principles for bootstrap measures and their joint distributions, enabling analysis of moderate deviations in bootstrap methods.

Findings

LDP for conditional distributions of bootstrap measures

LDP for joint distributions of empirical and bootstrap measures

Application to empirical quantile processes and copula functions

Abstract

We prove two Large deviations principles (LDP) in the zone of moderate deviation probabilities. First we establish LDP for the conditional distributions of moderate deviations of empirical bootstrap measures given empirical probability measures. Second we establish LDP for the joint distributions of empirical measure and bootstrap empirical measures. Using these LDPs, similar LDPs for statistical differentiable functionals can be established. The LDPs for moderate deviations of empirical quantile processes and empirical bootstrap copula function are provided as illustration of these results.