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Mathematical Methods of Statistics

, Volume 19, Issue 2, pp 121–135 | Cite as

Approximations for a multivariate hybrid process with applications to change-point detection

  • M. D. BurkeEmail author
Article

Abstract

We obtain probability inequalities and almost sure rates for the approximations of the hybrids of empirical and partial sums processes in the multivariate case. Applications to weighted bootstrap empirical processes as well as to change-point detection tests for general nonparametric regression models are discussed.

Key words

multivariate empirical processes partial sums change-point detection nonparametric regression rates of convergence 

2000 Mathematics Subject Classification

primary 60F17 secondary 62G08, 62F40 

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References

  1. 1.
    I. S. Borisov, “An Approximation of Empirical Fields”, in Nonparametric Statistical Inference, Coll. Math. Soc. János Bolyai, Budapest, Hungary, 1980, (North Holland, Amsterdam, 1982), Vol. 32, pp. 77–87.Google Scholar
  2. 2.
    I. S. Borisov, “An Approximation of Empirical Fields, Constructed with Respect to Vector Observations with Dependent Components”, Sibirsk. Mat. Zh. 23, 31–41 (1982).Google Scholar
  3. 3.
    M. D. Burke, “Multivariate Test-of-Fit and Uniform Confidence Bands Using a Weighted Bootstrap”, Statist. Probab. Letters 46, 13–20 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    M. D. Burke, “Limit Theorems for Change-Point Detection in Nonparametric Regression Models”, Acta Math. Sci. (Szeged) 73, 865–882 (2007).zbMATHMathSciNetGoogle Scholar
  5. 5.
    M. Csörgő and P. Révész, “A strong Approximation of the Multivariate Empirical Process”, Studia Sci. Math. Hung. 110, 427–434 (1975).Google Scholar
  6. 6.
    M. Csörgő and L. Horváth, “A Note on Strong Approximations of Multivariate Empirical Processes”, Stoch. Proc. Appl. 27, (1988) pp. 101–109.CrossRefGoogle Scholar
  7. 7.
    M. Csörgő and L. Horváth, Limit Theorems in Change-Point Analysis (Wiley, Chichester, 1997).Google Scholar
  8. 8.
    M. Csörgő, L. Horváth, and B. Szyszkowicz, “Integral Tests for Suprema of Kiefer Processes with Applications”, Statistics and Decisions 15, 365–377 (1998).Google Scholar
  9. 9.
    M. Csörgő and B. Szyszkowicz, “Weighted Multivariate Empirical Processes and Contiguous Change-Point Analysis”, In: Change-Point Problems, IMS Lecture Notes-Monograph Series (1994), Vol. 23, pp. 93–98.CrossRefGoogle Scholar
  10. 10.
    M. Csörgő and B. Szyszkowicz, “Applications of Multi-Time Parameter Processes to Change-Point Analysis”, in: Probability Theory and Mathematical Statistics (VSP/TEV, 1994), pp. 159–222.Google Scholar
  11. 11.
    J. Diebolt, “A Nonparametric Test for the Regression Function”, Asymptotic theory. J. Statist. Plan. Inference 44, 1–17 (1995).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    J. Komlós, P. Major, and G. Tusnády, “An Approximation of Partial Sums of Independent R.V.’s and the Sample Df. I.”, Z.Wahrsch. Verw. Geb. 32, 111–131 (1975).zbMATHCrossRefGoogle Scholar
  13. 13.
    L. Horváth, “Approximations for Hybrids of Empirical and Partial Sums Processes”, J. Statist. Plan. Inference 88, 1–18 (2000).zbMATHCrossRefGoogle Scholar
  14. 14.
    D. Y. Lin, J. M. Robins, and L. J. Wei, “Comparing Two Failure Time Distributions in the Presence of Dependent Censoring”, Biometrika 83, 381–393 (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    A. Y. Lo, “A Large Sample Study of the Bayesian Bootstrap”, Ann. Statist. 15, 360–375 (1987).zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    V. V. Petrov, Limit Theorems of Probability Theory: Sequences of Independent Random Variables (Oxford University Press, Oxford, 1995).zbMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2010

Authors and Affiliations

  1. 1.Dept. Math. and Statist.Univ. of CalgaryCalgaryCanada

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