Summary
The rate of convergence of the distribution function of a symmetric function of N independent and identically distributed random variables to its normal limit is investigated. Under appropriate moment conditions the rate is shown to be \(\mathcal{O}\)(N−1/2). This theorem generalizes many known results for special cases and two examples are given. Possible further extensions are indicated.
Article PDF
Similar content being viewed by others
References
Bickel, P.J.: Edgeworth expansions in nonparametric statistics. Ann. Statist. 2, 1–20 (1974)
Callaert, H., Janssen, P.: The Berry-Esseen theorem for U-statistics. Ann. Statist. 6, 417–421 (1978)
Chan, Y.-K., Wierman, J.: On the Berry-Esseen theorem for U-statistics, Ann. Probability 5, 136–139 (1977)
Efron, B., Stein, C.: The jackknife estimate of variance, Ann. Statist. 9, 586–596 (1981)
Feller, W.: An Introduction to Probability Theory and Its Applications. Vol. II, 2nd Ed. New York: Wiley 1971
Helmers, R.: A Berry-Esseen theorem for linear combinations of order statistics. Ann. Probability 9, 342–347 (1981)
Helmers, R.: Edgeworth Expansions for Linear Combinations of Order Statistics. Mathematical Centre Tracts 105. Mathematisch Centrum, Amsterdam (1982)
Helmers, R., Van Zwet, W.R.: The Berry-Esseen bound for U-statistics. Statistical Decision Theory and Related Topics, III Vol. 1, S.S. Gupta and J.O. Berger (eds.), 497–512. New York: Academic Press 1982
Hoeffding, W.: A class of statistics with asymptotically normal distributions. Ann. Math. Statist. 19, 293–325 (1948)
Hoeffding, W.: The strong law of large numbers for U-statistics. Inst. of Statist., Univ. of North Carolina, Mimeograph Series No. 302 (1961)
Karlin, S., Rinott, Y.: Applications of ANOVA type decompositions for comparisons of conditional variance statistics including jackknife estimates. Ann. Statist. 10, 485–501 (1982)
Author information
Authors and Affiliations
Additional information
Research supported by the U.S. Office of Naval Research, Contract N 00014-80-C-0163
Rights and permissions
About this article
Cite this article
van Zwet, W.R. A Berry-Esseen bound for symmetric statistics. Z. Wahrscheinlichkeitstheorie verw Gebiete 66, 425–440 (1984). https://doi.org/10.1007/BF00533707
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF00533707