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Random Quadratic Forms and the Bootstrap for U-Statistics

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  • Dehling, H.
  • Mikosch, T.

Abstract

We study the bootstrap distribution for U-statistics with special emphasis on the degenerate case. For the Efron bootstrap we give a short proof of the consistency using Mallows' metrics. We also study the i.i.d. weighted bootstrap [formula] where (Xi) and ([xi]i) are two i.i.d. sequences, independent of each other and where E[xi]i = 0, Var([xi]i) = 1. It turns out that, conditionally given (Xi), this random quadratic form converges weakly to a Wiener-Ito double stochastic integral [integral operator]10 [integral operator]10h(F-1(x), F-1(y)) dW(x) dW(y). As a by-product we get an a.s. limit theorem for the eigenvalues of the matrix Hn=((1/n)h(Xi, Xj))1

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:51:y:1994:i:2:p:392-413
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    Cited by:

    1. Tamara Fernández & Nicolás Rivera, 2021. "A reproducing kernel Hilbert space log‐rank test for the two‐sample problem," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(4), pages 1384-1432, December.
    2. Olimjon Sharipov & Martin Wendler, 2012. "Bootstrap for the sample mean and for -statistics of mixing and near-epoch dependent processes," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(2), pages 317-342.
    3. Tarik Bahraoui & Jean‐François Quessy, 2022. "Tests of multivariate copula exchangeability based on Lévy measures," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(3), pages 1215-1243, September.
    4. Michael H. Neumann, 2021. "Bootstrap for integer‐valued GARCH(p, q) processes," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 75(3), pages 343-363, August.
    5. Wang, Qiying & Jing, Bing-Yi, 2004. "Weighted bootstrap for U-statistics," Journal of Multivariate Analysis, Elsevier, vol. 91(2), pages 177-198, November.
    6. 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.
    7. Jiménez-Gamero, M. Dolores & Kim, Hyoung-Moon, 2015. "Fast goodness-of-fit tests based on the characteristic function," Computational Statistics & Data Analysis, Elsevier, vol. 89(C), pages 172-191.
    8. Dehling, Herold & Wendler, Martin, 2010. "Central limit theorem and the bootstrap for U-statistics of strongly mixing data," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 126-137, January.
    9. 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.
    10. M. D. Jiménez-Gamero & A. Batsidis, 2017. "Minimum distance estimators for count data based on the probability generating function with applications," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 80(5), pages 503-545, July.
    11. Jiménez-Gamero, M.D. & Alba-Fernández, M.V., 2021. "A test for the geometric distribution based on linear regression of order statistics," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 186(C), pages 103-123.
    12. Shuran Zhao & Xingzhong Xu & Xiaobo Ding, 2008. "The convergence rates of the weighted bootstrap distributions for von Mises and -statistics," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(7), pages 645-660.
    13. Dehling, Herold & Sharipov, Olimjon Sh. & Wendler, Martin, 2015. "Bootstrap for dependent Hilbert space-valued random variables with application to von Mises statistics," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 200-215.
    14. Quessy, Jean-François, 2021. "A Szekely–Rizzo inequality for testing general copula homogeneity hypotheses," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    15. Leucht, Anne & Neumann, Michael H. & Kreiss, Jens-Peter, 2013. "A model specification test for GARCH(1,1) processes," Working Papers 13-11, University of Mannheim, Department of Economics.
    16. Matsui, Muneya & Mikosch, Thomas & Roozegar, Rasool & Tafakori, Laleh, 2022. "Distance covariance for random fields," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 280-322.
    17. G. I. Rivas-Martínez & M. D. Jiménez-Gamero & J. L. Moreno-Rebollo, 2019. "A two-sample test for the error distribution in nonparametric regression based on the characteristic function," Statistical Papers, Springer, vol. 60(4), pages 1369-1395, August.
    18. Anne Leucht & Jens-Peter Kreiss & Michael H. Neumann, 2015. "A Model Specification Test For GARCH(1,1) Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(4), pages 1167-1193, December.
    19. Bartels, Knut, 1998. "A model specification test," SFB 373 Discussion Papers 1998,109, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    20. Leucht, Anne & Neumann, Michael H., 2009. "Consistency of general bootstrap methods for degenerate U-type and V-type statistics," Journal of Multivariate Analysis, Elsevier, vol. 100(8), pages 1622-1633, September.
    21. Bahraoui Tarik & Bouezmarni Taoufik & Quessy Jean-François, 2018. "Testing the symmetry of a dependence structure with a characteristic function," Dependence Modeling, De Gruyter, vol. 6(1), pages 331-355, December.
    22. Inass Soukarieh & Salim Bouzebda, 2022. "Exchangeably Weighted Bootstraps of General Markov U -Process," Mathematics, MDPI, vol. 10(20), pages 1-42, October.

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