Dependent wild bootstrap for degenerate U- and V-statistics

by    A. Leucht, M. H. Neumann

Preprint series: 11-02, Reports on Stochastics and Statistics

62M07 Non-Markovian processes: hypothesis testing
62G09 Resampling methods
62E20 Asymptotic distribution theory

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 approximate critical values of those 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 (2010, JASA 105, 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 the 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 is reasonably close to the nominal one, we see nontrivial empirical power in all cases considered.

Keywords: Bootstrap, weak dependence, U-statistic, V -statistic, Cramér-von Mises test, two-sample test

Upload: 2011 -12 -06

Update: 2011 -12 -06

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