Jackknife multiplier bootstrap: finite sample approximations to the $U$-process supremum with applications

TL;DR

This paper develops a novel jackknife multiplier bootstrap method for finite sample approximation of the supremum of non-degenerate U-processes, addressing challenges when the process is not weakly convergent and the covariance is unknown.

Contribution

It introduces a new bootstrap approach tailored for U-processes with changing function classes and distributions, providing non-asymptotic bounds and local maximal inequalities.

Findings

Proposes a jackknife multiplier bootstrap (JMB) for U-process supremum approximation.

Derives non-asymptotic coupling and Kolmogorov distance bounds for JMB.

Demonstrates applications in nonparametric function testing.

Abstract

This paper is concerned with finite sample approximations to the supremum of a non-degenerate $U$-process of a general order indexed by a function class. We are primarily interested in situations where the function class as well as the underlying distribution change with the sample size, and the $U$-process itself is not weakly convergent as a process. Such situations arise in a variety of modern statistical problems. We first consider Gaussian approximations, namely, approximate the $U$-process supremum by the supremum of a Gaussian process, and derive coupling and Kolmogorov distance bounds. Such Gaussian approximations are, however, not often directly applicable in statistical problems since the covariance function of the approximating Gaussian process is unknown. This motivates us to study bootstrap-type approximations to the $U$-process supremum. We propose a novel jackknife multiplier bootstrap (JMB) tailored to the $U$-process, and derive coupling and Kolmogorov distance bounds for the proposed JMB method. All these results are non-asymptotic, and established under fairly general conditions on function classes and underlying distributions. Key technical tools in the proofs are new local maximal inequalities for $U$-processes, which may be useful in other problems. We also discuss applications of the general approximation results to testing for qualitative features of nonparametric functions based on generalized local $U$-processes.