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

Xiaohui Chen, Kengo Kato

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)1097-1163
Number of pages67
JournalProbability Theory and Related Fields
Issue number3-4
StatePublished - Apr 1 2020


  • Coupling
  • Gaussian approximation
  • Jackknife multiplier bootstrap
  • Local maximal inequality
  • U-process

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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