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Symmetric Statistics

  • Rabi Bhattacharya
  • Manfred Denker
Chapter
Part of the DMV Seminar book series (OWS, volume 14)

Abstract

This chapter deals with principles of the asymptotic distribution theory for symmetric statistics, that are U-statistics, differentiable statistical functions and certain multiple stochastic integrals. Since the multiple sample case is delt with in a quite similar way, we can restrict attention to the classical one-sample statistics. General references are [D1], [RW] and [S1] among many other books. The first one contains some technical material needed to fill in details of the following discussion. It is also helpful to understand the little changes necessary in the multisample case.

Keywords

Invariance Principle Empirical Process Iterate Logarithm Empirical Distribution Function Symmetric Statistic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [BF]
    P. Bickel; D.A. Preedman: Some asymptotic theory for the bootstrap. Ann. Stat. 9, (1981), 1196–1217.CrossRefGoogle Scholar
  2. [BI]
    P. Billingsley: Convergence of probability measures. Wiley 1968.Google Scholar
  3. [CS]
    E. Csaki: On the standarized empirical distribution function. Coll. Math. Soc. Janos Bol. 32. Nonparametric statistical Inference. Budapest 1980, 123–138.Google Scholar
  4. [DH1]
    H. Dehling: The functional law of the iterated logarithm for von Mises functionals and multiple Wiener integrals. J. Mult. Anal. 28, (1989), 177–189.CrossRefGoogle Scholar
  5. [DH2]
    H. Dehling: Complete convergence of triangular arrays and the law of the iterated logarithm for U-statistics. Stat. & Probab. Letters 7, (1989), 319–321.CrossRefGoogle Scholar
  6. [DT]
    H. Dehling; M.S. Taqqu: The functional law of the iterated logarithm for the empirical process of some long range dependent sequences. Stat. & Probab. Letters, 7, (1989), 81–85.CrossRefGoogle Scholar
  7. [DDP1]
    H. Dehling; M. Denker; W. Philipp: Invariance principles for von Mises’- and U-statistics. Z. Wahrsch. Theorie verw. Geb. 67, (1984), 139–167.CrossRefGoogle Scholar
  8. [DDP2]
    H. Dehling; M. Denker; W. Philipp: A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U- statistics. Prob. Th. Rel. Fields, 72, (1986), 111–131.CrossRefGoogle Scholar
  9. [DDP2]
    H. Dehling; M. Denker; W. Philipp: The almost sure invariance principle for the empirical process of U-statistic structure. Ann. Inst. H. Poincaré, Prob. et Stat. 23, (1987), 121–134.Google Scholar
  10. [DDW]
    H. Dehling; M. Denker; W. Woyczynski: Resampling U-statistics using p-stable laws.Google Scholar
  11. [D1]
    M. Denker: Asymptotic distribution theory in nonparametric statistics. Vieweg 1985Google Scholar
  12. [DK]
    M. Denker; G. Keller: Rigorous statistical procedures for data from dynamical systems. J. Stat. Physics, 44, (1986), 67–93.CrossRefGoogle Scholar
  13. [DGK]
    M. Denker; C. Grillenberger; G. Keller: A note on invariance principles for v. Mises’ statistics. Metrika 32, (1985), 197–214.CrossRefGoogle Scholar
  14. [RWE]
    T. de Wet; R.H. Randies: On the effect of substituting parameter estimators in limiting χ2, U- and V-statistics. Ann. Stat. 15, (1987), 398–412.CrossRefGoogle Scholar
  15. [WV]
    T. de Wet; H.J. Venter: Asymptotic distributions for quadratic forms with applications to tests of fit. Ann. Stat. 1, (1973), 380–387.CrossRefGoogle Scholar
  16. [DM]
    E. Dynkin; A. Mandelbaum: Symmetric statistics, Poisson point processes and multiple Wiener integrals. Ann. Stat. 11, (1983), 739–745.CrossRefGoogle Scholar
  17. [EF]
    B. Efron: Bootstrap methods: another look at the jackknife. Ann. Stat. 7, (1979), 1–26.CrossRefGoogle Scholar
  18. [ERD]
    A. Erdely et al.: Higher transcendental functions 2. McGraw Hill, 1953.Google Scholar
  19. [FH]
    L.T. Fernholz: v. Mises calculus for statistical functional. Lect. Notes in Stat. 19, Springer 1983.Google Scholar
  20. [FI]
    A.A. Filippova: Mises’ theorem on the asymptotic behavior of functionals of empirical distribution function and its statistical applications. Theory Prob. Appl. 7, (1962), 24–57.CrossRefGoogle Scholar
  21. [GR]
    G.G. Gregory: Large sample theory for U-statistics and tests of fit. Ann. Stat. 5, (1977), 110–123.CrossRefGoogle Scholar
  22. [HA]
    P. Hall: On the invariance principle for U-statistics. Stoch. Proc. Appl. 9, (1979), 163–174.Google Scholar
  23. [H1]
    W. Hoeffding: A class of statistics with asymptotically normal distributions. Ann. Math. Stat. 19 19, (1948), 293–325.CrossRefGoogle Scholar
  24. [IT]
    K. Ito: Multiple Wiener integral. J. Math. Soc. Japan, 3, (1951), 157–164.CrossRefGoogle Scholar
  25. [MT]
    A. Mandelbaum; M.S. Taqqu: Invariance principles for symmetric statistics. Ann. Stat. 12, (1984), 483–496.CrossRefGoogle Scholar
  26. [MC]
    T. McConnell: Two parameter strong laws and maximal inequalities for U- statistics. Proc. Roy. Soc. Edinburgh 107A, (1987), 133–151.Google Scholar
  27. [vM]
    R. von Mises: On the asymptotic distribution of differentiable statistical functions. Ann. Math. Stat. 18, (1947), 309–348.CrossRefGoogle Scholar
  28. [MO]
    T. Mori; H. Oodaira: Freidlin-Wentzell type estimates and the law of the iterated logarithm for a class of stochastic processes related to symmetric statistics. Center for Stoch. Proc., Univ. N. Carolina, Chapel Hill, Tech. Rep. 184.Google Scholar
  29. [NE]
    G. Neuhaus: Functional limit theorems for U-statistics in the degenerate case. J. Mult. Anal. 7, (1977), 424–439.CrossRefGoogle Scholar
  30. [NP1]
    D. Nolan; D. Pollard: U-processes: rates of convergence. Ann. Stat. 15, (1987), 780–799.CrossRefGoogle Scholar
  31. [NP2]
    D. Nolan; D. Pollard: Functional limit theorems for U-processes. Ann. Prob. 16, (1988), 291–298.Google Scholar
  32. [PS]
    R. Pyke; G.R. Shorack: The weak convergence of the empirical process with random sample size. Proc. Cambridge Phil. Soc. 64, (1968), 155–160.CrossRefGoogle Scholar
  33. [RW]
    R.H. Randies; D.A. Wolfe: Introduction to the theory of nonparametric statistics. Wiley 1979.Google Scholar
  34. [S1]
    R.J. Serfling: Approximation theorems in mathematical statistics. Wiley 1980.CrossRefGoogle Scholar
  35. [S2]
    R.J. Serfling: Generalized L-, M- and R-statistics. Ann. Stat. 12, (1984), 76–86.CrossRefGoogle Scholar
  36. [SW]
    G.R. Shorack; J. Wellner: Empirical processes with applications to statistics. Wiley 1986.Google Scholar
  37. [SI]
    B.W. Silverman: Convergence of a class of empirical distribution functions of dependent random variables. Ann. Prob. 11, (1983), 745–751.CrossRefGoogle Scholar
  38. [WI]
    N. Wiener: The homogeneous chaos. Amer. J. Math. 60, (1930), 897–936.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel 1990

Authors and Affiliations

  • Rabi Bhattacharya
    • 1
  • Manfred Denker
    • 2
  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA
  2. 2.Institut für MathematischeStochastikGöttingenGermany

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