Empirical Processes and p-variation

  • R. M. Dudley
Part of the Selected Works in Probability and Statistics book series (SWPS)


Remainder bounds in Fréchet differentiability of functionals for p-variation norms are found for empirical distribution functions. For \(1 \le p \le 2\) the p-variation of the empirical process \(n^{1/2} (F_n - F)\) is of order \(n^{1 - p/2} \) in probability up to a factor (log log n)p/2. For \((F,G) \mapsto \smallint FdG\) and for \((F,G) \mapsto F \circ G^{ - 1} \) this yields nearly optimal remainder bounds. Also, p-variation gives new proofs for the asymptotic distributions of the Cramér-von Mises-Rosenblatt and Watson two-sample statistics when the two sample sizes m, n go to infinity arbitrarily.


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  1. Aki, S. (1981), ‘Asymptotic behavior of functionals of empirical distribution functions for the two-sample problem’, Annals of the Institute of Statistical Mathematics 33, 391–403.zbMATHCrossRefMathSciNetGoogle Scholar
  2. Andersen, P. K., Borgan, Ø, Gill, R. D. & Keiding, N. (1993), Statistical Models Based on Counting Processes, Springer-Verlag, New York.zbMATHGoogle Scholar
  3. Appell, J. & Zabrejko, P. P. (1990), Nonlinear Superposition Operators, Cambridge University Press.zbMATHGoogle Scholar
  4. Beirlant, J. & Deheuvels, P. (1990), ‘On the approximation of P-P and Q-Q plot processes by Brownian bridges’, Statistics and Probability Letters 9, 241–251.zbMATHCrossRefMathSciNetGoogle Scholar
  5. Darling, D. A. (1957), ‘The Kolmogorov-Smirnov, Cramér-von Mises tests’, Annals of Mathematical Statistics 28, 823–838.zbMATHCrossRefMathSciNetGoogle Scholar
  6. Dixon, W. J. (1940), ‘A criterion for testing the hypothesis that two samples are from the same population’, Annals of Mathematical Statistics 11, 199–204.zbMATHCrossRefMathSciNetGoogle Scholar
  7. Dudley, R. M. (1985), ‘An extended Wichura theorem, definitions of Donsker class, and weighted empirical distributions’, Springer Lecture Notes in Mathematics 1153, 141–178.CrossRefMathSciNetGoogle Scholar
  8. Dudley, R. M. (1992), ‘Fréchet differentiability, p-variation and uniform Donsker classes’, Annals of Probability 20, 1968–1982.zbMATHCrossRefMathSciNetGoogle Scholar
  9. Dudley, R. M. (1993), Real Analysis and Probability, Chapman and Hall, New York. Second printing, corrected.Google Scholar
  10. Dudley, R. M. (1994), ‘The order of the remainder in derivatives of composition and inverse operators for p-variation norms’, Annals of Statistics 22, 1–20.zbMATHCrossRefMathSciNetGoogle Scholar
  11. Fernholz, L. T. (1983), von Mises Calculus for Statistical Functionals, Vol. 19 of Lecture Notes in Statistics, Springer, New York.Google Scholar
  12. Filippova, A. A. (1961), ‘[the von] mises theorem on the asymptotic behavior of functionals of empirical distribution functions and its statistical applications’, Theory of Probability and Its Applications 7, 24–57.CrossRefGoogle Scholar
  13. Fisz, M. (1960), ‘On a result by M. Rosenblatt concerning the von Mises-Smirnov test’, Annals of Mathematical Statistics 31, 427–429.zbMATHCrossRefMathSciNetGoogle Scholar
  14. Gill, R. D. (1989), ‘Non- and semi-parametric maximum likelihood estimators and the von Mises method’, Scandinavian Journal of Statistics 16, 97–128.zbMATHMathSciNetGoogle Scholar
  15. Huang, Y.-C. (1994), Empirical distribution function statistics, speed of convergence, and p-variation, PhD thesis, Massachusetts Institute of Technology.Google Scholar
  16. Huang, Y.-C. (1995), Speed of convergence of classical empirical processes in p-variation norm, preprint, Academica Sinica, Taipei, Taiwan.Google Scholar
  17. Janson, S. (1984), ‘The asymptotic distributions of incomplete U-statistics’, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 66, 495–505.zbMATHCrossRefMathSciNetGoogle Scholar
  18. Kiefer, J.(1959), ‘K-sample analogues of the Kolmogorov-Smirnov and Cramér-v. Mises tests’, Annals of Mathematical Statistics 30, 420–447.zbMATHCrossRefMathSciNetGoogle Scholar
  19. Kiefer, J. (1970), Deviations between the sample quantile process and the sample df, in M. L. Puri, ed., ‘Nonparametric Techniques in Statistical Inference’, Cambridge University Press, pp. 299–319.Google Scholar
  20. Komlós, J., Major, P. & Tusnády, G. (1975), ‘An approximation of partial sums of independent RV’s, and the sample DF. I’, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 32, 111–131.zbMATHCrossRefGoogle Scholar
  21. Lehmann, E. L. (1951), ‘Consistency and unbiasedness of certain nonpara-metric tests’, Annals of Mathematical Statistics 22, 165–179.zbMATHCrossRefMathSciNetGoogle Scholar
  22. Persson, T. (1979), ‘A new way to obtain Watson’s U 2, Scandinavian Journal of Statistics 6, 119–122.zbMATHMathSciNetGoogle Scholar
  23. Randies, R. H. & Wolfe, D. A. (1991), Introduction to the Theory of Nonparametric Statistics, Krieger, Malabar, FL. Reprinted with corrections.Google Scholar
  24. Reeds, J. A. (1976), On the definition of von Mises functionals, PhD thesis, Statistics, Harvard University.Google Scholar
  25. Rosenblatt, M. (1952), ‘Limit theorems associated with variants of the von Mises statistic’, Annals of Mathematical Statistics 23, 617–623.zbMATHCrossRefMathSciNetGoogle Scholar
  26. Taylor, S. J. (1972), ‘Exact asymptotic estimates of Brownian path variation’, Duke Mathematical Journal 39, 219–241.zbMATHCrossRefMathSciNetGoogle Scholar
  27. Watson, G. S. (1962), ‘Goodness-of-fit tests on a circle, II’, Biometrika 49, 57–63.zbMATHMathSciNetGoogle Scholar
  28. Young, L. C. (1936), ‘An inequality of the Hölder type, connected with Stieltjes integration’, Acta Mathematica (Djursholm) 67, 251–282.zbMATHCrossRefGoogle Scholar
  29. Young, L. C. (1938), ‘General inequalities for Stieltjes integrals and the convergence of Fourier series’, Mathematische Annalen 115, 581–612.CrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • R. M. Dudley
    • 1
  1. 1.Mathematics DepartmentMassachusetts Institute of TechnologyCambridgeUS

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