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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.

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References

  • Anderson, T. W., Darling, D. A. (1954). A test of goodness of fit. Journal of the American Statistical Association, 49, 765–769.

    Google Scholar 

  • Andrews, D. W. K. (1984). Non-strong mixing autoregressive processes. Journal of Applied Probability, 21, 930–934.

    Google Scholar 

  • Arcones, M. A., Giné, E. (1992). On the bootstrap of \(U\) and \(V\) statistics. Annals of Statistics, 20, 655–674.

  • Babbel, B. (1989). Invariance principles for \(U\)-statistics and von Mises functionals. Journal of Statistical Planning and Inference, 22, 337–354.

    Google Scholar 

  • Bierens, H. J., Ploberger, W. (1997). Asymptotic theory of integrated conditional moment tests. Econometrica, 65, 1129–1151.

    Google Scholar 

  • Bierens, H. J., Wang, L. (2012). Integrated conditional moment tests for parametric conditional distributions. Econometric Theory, 28, 328–362.

    Google Scholar 

  • Billingsley, P. (1999). Convergence of Probability Measures (2nd ed.). New York: Wiley.

  • Borisov, I.S., Bystrov, A. A. (2006). Limit theorems for the canonical von Mises statistics with dependent data. Sibirski Matematicheski Zhurnal 47, 1205–1217 (English translation: Siberian Mathematical Journal, 47, 980–989.).

    Google Scholar 

  • Borisov, I. S., Volodko, N. V. (2008). Orthogonal series and limit theorems for canonical \(U\)- and \(V\)-statistics of stationary connected observations. Siberian Advances in Mathematics, 18, 242–257.

    Google Scholar 

  • Bradley, R. C. (2007). Introduction to Strong Mixing Conditions (Vol. I). Heber City: Kendrick Press.

  • Carlstein, E. (1988). Degenerate \(U\)-statistics based on non-independent observations. Calcutta Statistical Association Bulletin, 37, 55–65.

    Google Scholar 

  • Davis, R. A., Dunsmuir, W. T. M., Streett, S. B. (2003). Observation-driven models for Poisson counts. Biometrika, 90, 777–790.

    Google Scholar 

  • Dehling, H., Mikosch, T. (1994). Random quadratic forms and the bootstrap for \(U\)-statistics. Journal of Multivariate Analysis, 51, 392–413.

    Google Scholar 

  • Dehling, H., Taqqu, M. S. (1991). Bivariate symmetric statistics of long-range dependent observations. Journal of Statistical Planning and Inference, 28, 153–165.

    Google Scholar 

  • Dewan, I., Prakasa Rao, B. L. S. (2001). Asymptotic normality for \(U\)-statistics of associated random variables. Journal of Statistical Planning and Inference, 97, 201–225.

    Google Scholar 

  • de Wet, T. (1987). Degenerate \(U\)- and \(V\)-statistics. South African Statistical Journal, 21, 99–129.

  • de Wet, T., Randles, R. H. (1987). On the effect of substituting parameter estimators in limiting \(\chi ^2 U\) and \(V\) statistics. Annals of Statistics, 15, 398–412.

  • de Wet, T., Venter, J. H. (1973). Asymptotic distributions for quadratic forms with applications to tests of fit. Annals of Statistics, 1, 380–387.

    Google Scholar 

  • Dudley, R. M. (1989). Real Analysis and Probability. New York: Chapman and Hall.

  • Dunford, N., Schwartz, J. T. (1963). Linear operators, part II, spectral theory. Self adjoint operators in Hilbert space. New York: John Wiley and Sons.

  • Dynkin, E. B., Mandelbaum, A. (1983). Symmetric statistics, Poisson point processes, and multiple Wiener integrals. Annals of Statistics, 11, 739–745.

    Google Scholar 

  • Eagleson, G. K. (1979). Orthogonal expansions and \(U\)-statistics. Australian Journal of Statistics, 21, 221–237.

    Google Scholar 

  • Escanciano, J. C. (2007). Model checks using residual marked empirical processes. Statistica Sinica, 17, 115–138.

    Google Scholar 

  • Fan, Y., Li, Q. (1999). Central limit theorem for degenerate \(U\)-statistics of absolutely regular processes with applications to model specification testing. Journal of Nonparametric Statistics, 10, 245–271.

    Google Scholar 

  • Fan, Y., Li, Q. (2000). Consistent model specification tests: kernel-based tests versus Bierens’ ICM tests. Econometric Theory, 16, 1016–1041.

    Google Scholar 

  • Ferland, R., Latour, A., Oraichi, D. (2006). Integer-valued GARCH processes. Journal of Time Series Analysis, 27, 923–942.

    Google Scholar 

  • Fokianos, K., Neumann, M. H. (2012). A goodness-of-fit test for Poisson count processes (Preprint).

  • Fokianos, K., Tjøstheim, D. (2011). Log-linear Poisson autoregression. Journal of Multivariate Analysis, 102, 563–578.

    Google Scholar 

  • Fokianos, K., Rahbek, A., Tjøstheim, D. (2009). Poisson autoregression. Journal of the American Statistical Association, 104, 1430–1439.

    Google Scholar 

  • Gao, J., King, M., Lu, Z., Tjøstheim, D. (2009). Specification testing in nonlinear and nonstationary time series regression. Annals of Statistics, 37, 3893–3928.

    Google Scholar 

  • Gregory, G. G. (1977). Large sample theory for \(U\)-statistics and tests of fit. Annals of Statistics, 5, 110–123.

    Google Scholar 

  • Huang, W., Zhang, L.-X. (2006). Asymptotic normality for \(U\)-statistics of negatively associated random variables. Statistics & Probability Letters, 76, 1125–1131.

    Google Scholar 

  • Leucht, A. (2010). Consistent model-specification tests based on parametric bootstrap. Reports of the Department of Mathematics and Computer Science, Friedrich Schiller University Jena 10-07.

  • Leucht, A. (2012). Degenerate \(U\)- and \(V\)-statistics under weak dependence: Asymptotic theory and bootstrap consistency. Bernoulli, 18, 552–585.

  • Leucht, A., Neumann, M. H. (2009). Consistency of general bootstrap methods for degenerate \(U\)- and \(V\)-type statistics. Journal of Multivariate Analysis, 100, 1622–1633.

    Google Scholar 

  • Leucht, A., Neumann, M. H. (2011). Degenerate \(U\)- and \(V\)-statistics under ergodicity: Asymptotics, bootstrap and applications in statistics. Reports of the Department of Mathematics and Computer Science, Friedrich Schiller University Jena 11-01.

  • McLeish, D. L. (1974). Dependent central limit theorems and invariance principles. Annals of Probability, 2, 620–628.

    Google Scholar 

  • Neumann, M. H. (2011). Absolute regularity and ergodicity of Poisson count processes. Bernoulli, 17, 1268–1284.

    Google Scholar 

  • Neumann, M. H., Paparoditis, E. (2008). Goodness-of-fit tests for Markovian time series models: Central limit theory and bootstrap approximations. Bernoulli, 14, 14–46.

    Google Scholar 

  • R Development Core Team (2007). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, http://www.R-project.org.

  • Rydberg, T. H., Shephard, N. (2000). BIN models for trade-by-trade data. In Modelling the number of trades in a fixed interval of time. World Conference Econometric Society, 2000, Seattle, Contributed Paper 0740.

  • Serfling, R. S. (1980). Approximation Theorems of Mathematical Statistics. New York: Wiley.

  • Stinchcombe, M. B., White, H. (1998). Consistent specification testing with nuisance parameters present only under the alternative. Econometric Theory, 14, 295–325.

    Google Scholar 

  • Streett, S. (2000). Some observation driven models for time series of counts. Ph.D. Thesis, Colorado State University, Department of Statistics, Fort Collins.

  • Sun, H. (2005). Mercer theorem for RKHS on noncompact sets. Journal of Complexity, 214, 337–349.

    Google Scholar 

  • van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge: Cambridge University Press.

Download references

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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Correspondence to Anne Leucht.

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Leucht, A., Neumann, M.H. Degenerate \(U\)- and \(V\)-statistics under ergodicity: asymptotics, bootstrap and applications in statistics. Ann Inst Stat Math 65, 349–386 (2013). https://doi.org/10.1007/s10463-012-0374-9

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  • DOI: https://doi.org/10.1007/s10463-012-0374-9

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