Random quadratic forms and the bootstrap for U-statistics
Journal of Multivariate Analysis
A test for the two-sample problem based on empirical characteristic functions
Computational Statistics & Data Analysis
Mercer theorem for RKHS on noncompact sets
Journal of Complexity
Characteristic function-based hypothesis tests under weak dependence
Journal of Multivariate Analysis
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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 Cramer-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 Cramer-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.