Probability Theory and Related Fields

, Volume 79, Issue 1, pp 75–93

Glivenko-Cantelli properties of some generalized empirical DF's and strong convergence of generalized L-statistics

  • R. Helmers
  • P. Janssen
  • R. Serfling
Article

Summary

We study a nonclassical form of empirical df Hnwhich is of U-statistic structure and extend to Hnthe classical exponential probability inequalities and Glivenko-Cantelli convergence properties known for the usual empirical df. An important class of statistics is given byT(Hn), where T(·) is a generalized form of L-functional. For such statisticswe prove almost sure convergence using an approach which separates the functional-analytic and stochastic components of the problem and handles the latter component by application of Glivenko-Cantelli type properties.Classical results for U-statistics and L-statistics are obtained as special cases without addition of unnecessary restrictions.Many important new types of statistics of current interest are covered as well by our result.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bickel, P.J., Lehmann, E.L.: Descriptive statistics for non-parametric models. IV. Spread. In: Jurečková, J. (ed.) Contributions to statistics. Hájek Memorial Volume, pp. 33–40. Prague: Academia 1979Google Scholar
  2. Chung, K.L.: An estimate concerning the Kolmogorov limit distribution. Trans. Am. Math. Soc. 67, 36–50 (1949)Google Scholar
  3. Dunford, N., Schwartz, J.T.: Linear operators, vol. I. New York: Wiley 1958Google Scholar
  4. Dvoretzky, A., Kiefer, J., Wolfowitz, J.: Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann. Math. Statist. 27, 642–669 (1956)Google Scholar
  5. Gaenssler, P.: Empirical processes, Hayward, Calif.: Institute of Mathematical Statistics, Lecture Notes-Monograph Series, vol. 3 (1983)Google Scholar
  6. Helmers, R.: Edgeworth expansions for linear combinations of order statistics. Mathematical Centre Tracts, vol. 105. Amsterdam: Mathematisch Centrum 1982Google Scholar
  7. Helmers, R., Janssen, P., Serfling, R.: Glivenko-Cantelli properties of some generalized empirical df's and strong convergence of generalized L-statistics. Technical Report No. 460, Dept. of Math. Sciences, Johns Hopkins University, Baltimore (1985)Google Scholar
  8. Hoeffding, W.: The strong law of large numbers for U-statistics. Univ. of North Carolina Institute of Statistics Mimeo Series, No. 302 (1961)Google Scholar
  9. Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58, 13–30 (1963)Google Scholar
  10. Janssen, P., Serfling, R., Veraverbeke, N.: Asymptotic normality for a general class of statistical functions and application to measure of spread. Ann. Stat. 12, 1369–1379 (1984)Google Scholar
  11. McConnell, T.R.: Two-parameter strong laws and maximal inequalities for U-statistics. Proc. of the Royal Soc. of Edinburgh 107A, 133–151 (1987)Google Scholar
  12. Sen, P.K.: Almost sure convergence of generalized U-statistics. Ann. Probab. 5, 287–290 (1977)Google Scholar
  13. Serfling, R.J.: Approximation theorems of mathematical statistics. New York: Wiley 1980Google Scholar
  14. Serfling, R.J.: Generalized L-, M-and R-statistics. Ann. Stat. 12, 76–86 (1984)Google Scholar
  15. Serfling, R.J.: A bahadur representation for quantiles of empirical df's of generalized U-statistic structure. Technical Report No. 453, Dept. of Math. Sciences, Johns Hopkins University, Baltimore (1985a)Google Scholar
  16. Serfling, R.J.: A note on convergence of functions of random elements. Technical Report No. 459, Dept. of Math. Sciences, Johns Hopkins University, Baltimore (1985b)Google Scholar
  17. van Zwet, W. R.: A strong law for linear functions of order statistics. Ann. Probab. 8, 986–990 (1980)Google Scholar
  18. Wellner, J.A.: A Glivenko-Cantelli theorem and strong laws of large numbers for functions of order statistics. Ann. Stat. 5, 473–480 (1977). Correction, ibid. 6, 1394 (1977)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • R. Helmers
    • 1
  • P. Janssen
    • 2
  • R. Serfling
    • 3
  1. 1.Centre for Mathematics and Computer ScienceAmsterdamThe Netherlands
  2. 2.Limburgs Universitair CentrumDiepenbeekBelgium
  3. 3.Department of Mathematical SciencesJohns Hopkins UniversityBaltimoreUSA

Personalised recommendations