Dependent wild bootstrap for degenerate UU- and VV-statistics

Degenerate UU- and VV-statistics play an important role in the field of hypothesis testing since numerous test statistics can be formulated in terms of these quantities. Therefore, consistent bootstrap methods for UU- and VV-statistics can be applied in order to determine critical values for these tests. We prove a new asymptotic result for degenerate UU- and VV-statistics of weakly dependent random variables. As our main contribution, we propose a new model-free bootstrap method for UU- and VV-statistics of dependent random variables. Our method is a modification of the dependent wild bootstrap recently proposed by Shao [X. Shao, The dependent wild bootstrap, J. Amer. Statist. Assoc. 105 (2010) 218–235], where we do not directly bootstrap the underlying random variables but the summands of the UU- and VV-statistics. Asymptotic theory for the original and bootstrap statistics is derived under simple and easily verifiable conditions. We discuss applications to a Cramér–von Mises-type test and a two sample test for the marginal distribution of a time series in detail. The finite sample behavior of the Cramér–von Mises test is explored in a small simulation study. While the empirical size was reasonably close to the nominal one, we obtained nontrivial empirical power in all cases considered.