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Dependent wild bootstrap for degenerate U- and V-statistics

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  • Leucht, Anne
  • Neumann, Michael H.

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.

Suggested Citation

  • Leucht, Anne & Neumann, Michael H., 2013. "Dependent wild bootstrap for degenerate U- and V-statistics," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 257-280.
    • Handle: RePEc:eee:jmvana:v:117:y:2013:i:c:p:257-280
    DOI: 10.1016/j.jmva.2013.03.003
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    References listed on IDEAS

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    1. Neumann, Michael H. & Paparoditis, Efstathios, 2000. "On bootstrapping L2-type statistics in density testing," Statistics & Probability Letters, Elsevier, vol. 50(2), pages 137-147, November.
    2. Dehling, Herold & Durieu, Olivier & Volny, Dalibor, 2009. "New techniques for empirical processes of dependent data," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3699-3718, October.
    3. Anne Leucht & Michael Neumann, 2013. "Degenerate $$U$$ - and $$V$$ -statistics under ergodicity: asymptotics, bootstrap and applications in statistics," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(2), pages 349-386, April.
    4. Shao, Xiaofeng, 2010. "The Dependent Wild Bootstrap," Journal of the American Statistical Association, American Statistical Association, vol. 105(489), pages 218-235.
    5. Axel Munk & Jean-Pierre Stockis & Janis Valeinis & Götz Giese, 2011. "Neyman smooth goodness-of-fit tests for the marginal distribution of dependent data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(5), pages 939-959, October.
    6. Leucht, Anne, 2012. "Characteristic function-based hypothesis tests under weak dependence," Journal of Multivariate Analysis, Elsevier, vol. 108(C), pages 67-89.
    7. Dehling, H. & Mikosch, T., 1994. "Random Quadratic Forms and the Bootstrap for U-Statistics," Journal of Multivariate Analysis, Elsevier, vol. 51(2), pages 392-413, November.
    8. Anderson, N. H. & Hall, P. & Titterington, D. M., 1994. "Two-Sample Test Statistics for Measuring Discrepancies Between Two Multivariate Probability Density Functions Using Kernel-Based Density Estimates," Journal of Multivariate Analysis, Elsevier, vol. 50(1), pages 41-54, July.
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    4. Smeekes, S. & Urbain, J.R.Y.J., 2014. "A multivariate invariance principle for modified wild bootstrap methods with an application to unit root testing," Research Memorandum 008, Maastricht University, Graduate School of Business and Economics (GSBE).
    5. Doukhan, Paul & Lang, Gabriel & Leucht, Anne & Neumann, Michael H., 2014. "Dependent wild bootstrap for the empirical process," Working Papers 35246, University of Mannheim, Department of Economics.
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