The Annals of Statistics

Empirical Edgeworth expansions for symmetric statistics

Hein Putter and Willem R. van Zwet

Full-text: Open access


In this paper the validity of a one-term Edgeworth expansion for Studentized symmetric statistics is proved. We propose jackknife estimates for the unknown constants appearing in the expansion and prove their consistency. As a result we obtain the second-order correctness of the empirical Edgeworth expansion for a very general class of statistics, including $U$-statistics, $L$-statistics and smooth functions of the sample mean. We illustrate the application of the bootstrap in the case of a $U$-statistic of degree two.

Article information

Ann. Statist., Volume 26, Number 4 (1998), 1540-1569.

First available in Project Euclid: 21 June 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory 62G09: Resampling methods

Edgeworth expansion Hoeffding’s decomposition jackknife boot-strap


Putter, Hein; van Zwet, Willem R. Empirical Edgeworth expansions for symmetric statistics. Ann. Statist. 26 (1998), no. 4, 1540--1569. doi:10.1214/aos/1024691253.

Export citation


  • BENTKUS, V., GOTZE, F. and VAN ZWET, W. R. 1997. An Edgeworth expansion for sy mmetric ¨ statistics. Ann. Statist. 25 851 896. Z.
  • BICKEL, P. J., GOTZE, F. and VAN ZWET, W. R. 1986. The Edgeworth expansion for U-statistics of ¨ degree 2. Ann. Statist. 14 1463 1484.
  • CALLAERT, H. and VERAVERBEKE, N. 1981. The order of the normal approximation for a Studentized U-statistic. Ann. Statist. 9 194 200. Z.
  • DHARMADHIKARI, S. W., FABIAN, V. and JOGDEO, K. 1968. Bounds on the moments of martingales. Ann. Math. Statist. 39 1719 1723. Z.
  • HELMERS, R. 1991. On the Edgeworth expansion and the bootstrap approximation for a Studentized U-statistic. Ann. Statist. 19 470 484. Z.
  • HOEFFDING, W. 1948. A class of statistics with asy mptotically normal distribution. Ann. Math. Statist. 19 293 325. Z.
  • JANSSEN, P. 1978. The Berry Esseen theorem and an asy mptotic expansion for U-statistics, Z. Ph.D. thesis, Limburg Univ. Centre in Dutch. Z.
  • PUTTER, H. 1994. Consistency of resampling methods. Ph.D. thesis, Univ. Leiden. Z.
  • QUENOUILLE, M. H. 1949. Approximate tests of correlation in time series. J. Roy. Statist. Soc. Ser. B 11 68 84. Z.
  • QUENOUILLE, M. H. 1956. Notes on bias in estimation. Biometrika 43 353 360. Z.
  • TUKEY, J. W. 1958. Bias and confidence in not-quite large samples. Ann. Math. Statist. 29 614 Z. abstract. Z.
  • VAN ZWET, W. R. 1984. A Berry Esseen bound for sy mmetric statistics. Z. Wahrsch. Verw. Gebiete 66 425 440. Z.
  • VON BAHR, B. and ESSEEN C. G. 1965. Inequalities for the rth absolute moment of a sum of random variables, 1 r 2. Ann. Math. Statist. 36 299 303.