Abstract
In this paper we obtain asymptotic representations of several variance estimators of U-statistics and study their effects for studentizations via Edgeworth expansions. Jackknife, unbiased and Sen's variance estimators are investigated up to the order op(n-1). Substituting these estimators to studentized U-statistics, the Edgeworth expansions with remainder term o(n-1) are established and inverting the expansions, the effects on confidence intervals are discussed theoretically. We also show that Hinkley's corrected jackknife variance estimator is asymptotically equivalent to the unbiased variance estimator up to the order op(n-1).
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References
Arvesen, J. N. (1969). Jackknifing U-statistics, Ann. Math. Statist., 40, 2076–2100.
Bickel, P. J., Goetze, F. and van Zwet W. R. (1986). The Edgeworth expansion for U-statistics of degree two, 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.
Callaert, H., Janssen, P. and Veraverbeke, N. (1980). An Edgeworth expansion for U-statistics, Ann. Statist., 8, 299–312.
Dharmadhikari, S. W., Fabian, V. and Jogdeo, K. (1968). Bounds on the moments of martingales, Ann. Math. Statist., 39, 1719–1723.
Efron, B. and Stein, C. (1981). The jackknife estimate of variance, Ann. Statist., 9, 586–596.
Hall, P. (1992). The Bootstrap and Edgeworth Expansion, Springer, New York.
Helmers, R. (1991). On the Edgeworth expansion and the bootstrap approximation for a studentized U-statistic, Ann. Statist., 19, 470–484.
Hinkley, D. V. (1978). Improving the jackknife with special reference to correlation estimation, Biometrika, 65, 13–21.
Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution, Ann. Math. Statist., 19, 293–325.
Lai, T. L. and Wang, J. Q. (1993). Edgeworth expansion for symmetric statistics with applications to bootstrap methods, Statistica Sinica, 3, 517–542.
Maesono, Y. (1994). Edgeworth expansions of a studentized U-statistic and a jackknife estimator of variance, The Australian National University Statistics Research Report, No. SRR 038–94.
Maesono, Y. (1995). On the normal approximations of studentized U-statistic, J. Japan Statist. Soc., 25, 19–33.
Schucany, W. R. and Bankson, D. M. (1989). Small sample variance estimators for U-statistics, Austral. J. Statist., 31, 417–426.
Sen, P. K. (1960). On some convergence properties of U-statistics, Calcutta Statist. Assoc. Bull., 10, 1–18.
Sen, P. K. (1977). Some invariance principles relating to jackknifing and their role in sequential analysis, Ann. Statist., 5, 316–329.
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Maesono, Y. Asymptotic Comparisons of Several Variance Estimators and their Effects for Studentizations. Annals of the Institute of Statistical Mathematics 50, 451–470 (1998). https://doi.org/10.1023/A:1003521327411
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DOI: https://doi.org/10.1023/A:1003521327411