Dependent multiplier bootstraps for non-degenerate $U$-statistics under mixing conditions with applications

Abstract

The asymptotic validity of a resampling method for two sequential processes constructed from non-degenerate $U$-statistics is established under mixing conditions. The resampling schemes, referred to as {\em dependent multiplier bootstraps}, result from an adaptation of the seminal approach of \cite{GomHor02} to mixing sequences. The proofs exploit recent results of \cite{DehWen10b} on degenerate $U$-statistics. A data-driven procedure for estimating a key bandwidth parameter involved in the resampling schemes is also suggested, making the use of the studied dependent multiplier bootstraps fully automatic. The derived results are applied to the construction of confidence intervals and to test for change-point detection. For such applications, Monte Carlo experiments suggest that the use of the proposed resampling approaches can have advantages over that of estimated asymptotic distributions.