Metrika

, Volume 32, Issue 1, pp 351–362 | Cite as

Rate of convergence to normality forU-statistics with kernel of arbitrary degree

  • M. Aerts
Publication
  • 27 Downloads

Summary

A recent result ofHelmers/van Zwet [1981] concerning the rate of convergence to normality forU-statistics with kernel of degree two is extended toU-statistics with kernel of arbitrary degree. At the same time, this study includes the case where, for 0<δ≤1:
$$E|h (X_1 ,...,X_r )|^p< \infty ,p > \frac{{4 + \delta }}{3},and E|g (X_1 )|^{2 + \delta }< \infty .$$

Keywords

Stochastic Process Probability Theory Economic Theory Recent Result Arbitrary Degree 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bickel, P.J.: Edgeworth expansions in nonparameter statistics. Ann. Statist.2, 1974, 1–20.Google Scholar
  2. Callaert, H., andP. Janssen: The Berry-Esseen theorem forU-statistics. Ann. Statist.6, 1978, 417–421.Google Scholar
  3. —: A note on the convergence rate of random sums. Revue Roumaine de Math. Pures et Appl.28, 1983, 147–151.Google Scholar
  4. Chan, Y.-K., andJ. Wierman: On the Berry-Esséen theorem forU-statistics. Ann. Probability5, 1977, 136–139.Google Scholar
  5. Chatterji, S.D.: AnL p-convergence theorem. Ann. Math. Statist.40, 1969, 1068–1070.Google Scholar
  6. Dharmadhikari, S.W., V. Fabian andK. Jogdeo: Ann. Math. Statist.39, 1968, 1719–1723.Google Scholar
  7. Esséen, C.G.: Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian Law. Acta Math.77, 1945, 1–125.Google Scholar
  8. Feller, W.: An introduction to probability theory and its applications. II. J. Wiley, New-York 1966.Google Scholar
  9. Grams, W.F., andR.J. Serfling: Convergence rates forU-statistics and related statistics. Ann. Statist.1, 1973, 153–160.Google Scholar
  10. Helmers, R., andW.R. van Zwet: The Berry-Esséen bound forU-statistics. Statistical theory and related topics III. Ed. S.S. Gupta. Proceedings of a symposium held at Purdue University, 1981.Google Scholar
  11. Hoeffding, W.: A class of statistics with asymptotically normal distribution. Ann. Math. Statist.19, 1948, 293–325.Google Scholar
  12. —: The strong law of large numbers forU-statistics. Institute of Statistics Mimeo Series No.302, University of North Carolina, Chapel Hill, N.C. 1961.Google Scholar
  13. Janssen, P.: Rate of Convergence in the Central Limit Theorem and in the Strong Law of Large Numbers for von Mises Statistics. Metrika28, 1981, 35–46.Google Scholar
  14. Loève, M.: Probability Theory I, 4th ed., Springer-Verlag, New York 1977.Google Scholar
  15. Sproule, R.N.: Asymptotic properties ofU-statistics. Transactions. Am. Math. Soc.199, 1974, 55–64.Google Scholar

Copyright information

© Physica-Verlag Ges.m.b.H. 1985

Authors and Affiliations

  • M. Aerts
    • 1
  1. 1.Department of MathematicsLimburgs, Universitair CentrumDiepenbeekBelgium

Personalised recommendations