Source: Ann. Statist. Volume 26, Number 4
(1998), 1540-1569.
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.
References
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.
Mathematical Reviews (MathSciNet):
MR600547
DHARMADHIKARI, S. W., FABIAN, V. and JOGDEO, K. 1968. Bounds on the moments of martingales. Ann. Math. Statist. 39 1719 1723. Z.
Mathematical Reviews (MathSciNet):
MR230363
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.
Mathematical Reviews (MathSciNet):
MR26294
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.
Mathematical Reviews (MathSciNet):
MR751580
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.
Mathematical Reviews (MathSciNet):
MR170407
