Abstract
An Edgeworth expansion with remainder o(N−1) is established for a U-statistic with a kernel h of degree 2. The assumptions involved appear to be very mild; in particular, the common distribution of the summands h(X i , X j ) is not assumed to be smooth.
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Received J anuary 19 85; revised November 1985.
1 Research supported by the U.S. Office of Naval Research, Contract N0014-80-C-1063 a nd by the Netherlands Organization for Pure Scientific Research.
2 Research supported by the U.S. Office of Naval Research, Contract N00014-80-C-l063.
AMS 1980 subject classifications. Primary 62E20; secondary 60F05.
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ALBERS, W., BICKEL, P. J. and VAN ZWET, W. R. (1976). Asymptotic expansions for the power of distribution-free tests in t he one-sample problem. Ann. Statist. 4 108- 156.
BICKEL, P. J. (1974). Edgeworth expansions in nonparametric statistics. Ann. Statist. 2 1-20.
CALLAERT, H. and JANSSEN, P. (1978). The Berry-Esseen theorem for U-statistics. Ann. Statist. 6 417- 421.
CALLAERT, H., JANSSEN, P. and VERA VERBEKE , N. (1980). An Edgeworth expansion for U-statistics. Ann. Statist. 8 299- 312.
CHAN, Y.-K. and WIERMAN, J. (1977). On the Berry-Esseen theorem for U-statistics. Ann. Probab. 5 136-139.
DHARMADHIKARI, S. W., FABIAN, V. and JOGDEO, K. (1968). Hounds on t he moments of martingales. Ann. Math. Statist. 39 1719-1723.
FELLER, W. (1 971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York.
GoTZE, F . (1979). Asymptotic expansions for bivariate von Mises functionals. Z. Wahrsch. verw. Gebie te 50 333- 355.
HELMERS, R. and VAN ZWET, W. R. (1982). The Berry-Esseen bound for U-statistics. Statistical Decision Theory and R elated Topics, III (S. S. Gupta and J. 0. B e rger, eds. ) 1 497-512. Academic, New York.
HOEFFDING, W. (1948). A class of statistics with asymptotically normal distribution s. Ann. Math . Statist. 19 293- 325.
JANSSEN, P. L . (1978). D e Berry- Esseen Stelling en een Asymptotisc he Ontwikkeling voor U -statistieken. Ph.D. thesis, Dept. of Mathematics, Catholic Univ. of Leuven.
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Bickel, P.J., Götze, F., van Zwet, W.R. (2012). The Edgeworth Expansion for U-Statistics of Degree Two. In: van de Geer, S., Wegkamp, M. (eds) Selected Works of Willem van Zwet. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1314-1_14
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