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This paper is concerned with finite sample approximations to the supremum ofa non-degenerate $U$-process of a general order indexed by a function class. Weare primarily interested in situations where the function class as well as theunderlying distribution change with the sample size, and the $U$-process itselfis not weakly convergent as a process. Such situations arise in a variety ofmodern statistical problems. We first consider Gaussian approximations, namely,approximate the $U$-process supremum by the supremum of a Gaussian process, andderive coupling and Kolmogorov distance bounds. Such Gaussian approximationsare, however, not often directly applicable in statistical problems since thecovariance function of the approximating Gaussian process is unknown. Thismotivates us to study bootstrap-type approximations to the $U$-processsupremum. We propose a novel jackknife multiplier bootstrap (JMB) tailored tothe $U$-process, and derive coupling and Kolmogorov distance bounds for theproposed JMB method. All these results are non-asymptotic, and establishedunder fairly general conditions on function classes and underlyingdistributions. Key technical tools in the proofs are new local maximalinequalities for $U$-processes, which may be useful in other problems. We alsodiscuss applications of the general approximation results to testing forqualitative features of nonparametric functions based on generalized local$U$-processes.

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