Dependent wild bootstrap for degenerate U- and V-statistics

Anne Leucht and Michael H. Neumann

Journal of Multivariate Analysis, 2013, vol. 117, issue C, 257-280

Abstract: Degenerate U- and V-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 U- and V-statistics can be applied in order to determine critical values for these tests. We prove a new asymptotic result for degenerate U- and V-statistics of weakly dependent random variables. As our main contribution, we propose a new model-free bootstrap method for U- and V-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 U- and V-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.

Keywords: Bootstrap; Weak dependence; U-statistic; V-statistic; Cramér–von Mises test; Two-sample test (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (16)

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DOI: 10.1016/j.jmva.2013.03.003

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