Degenerate \(U\)- and \(V\)-statistics under ergodicity: asymptotics, bootstrap and applications in statistics

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Abstract

We derive the asymptotic distributions of degenerate \(U\)- and \(V\)-statistics of stationary and ergodic random variables. Statistics of these types naturally appear as approximations of test statistics. Since the limit variables are of complicated structure, typically depending on unknown parameters, quantiles can hardly be obtained directly. Therefore, we prove a general result on the consistency of model-based bootstrap methods for \(U\)- and \(V\)-statistics under easily verifiable conditions. Three applications to hypothesis testing are presented. Finally, the finite sample behavior of the bootstrap-based tests is illustrated by a simulation study.

Keywords

Bootstrap Ergodicity \(U\)-statistic \(V\)-statistic Cramér-von Mises-type test 
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Notes

Acknowledgments

This research was funded by the German Research Foundation DFG, projects NE 606/2-1 and NE 606/2-2.

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Copyright information

© The Institute of Statistical Mathematics, Tokyo 2012

Authors and Affiliations

  1. 1.Fachbereich Mathematik, SPSTUniversität HamburgHamburgGermany
  2. 2.Institut für StochastikFriedrich-Schiller-Universität JenaJenaGermany

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