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Strong laws for randomly indexed U-statistics

Published online by Cambridge University Press:  24 October 2008

Lajos Horváth
Lajos Horváth
Affiliation:
Bolyai Institute, Szeged University, H-6720 Szeged, Hungary and Department of Mathematics and Statistics, Carleton University, Ottawa K1S 5B6, Canada
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Abstract

A strong approximation for randomly indexed U-statistics is derived. The special case when the indices are stopping times based upon the jackknife estimator of the variance is discussed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 98 , Issue 3 , November 1985 , pp. 559 - 567
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

REFERENCES

[1]Aerts, M. and Callaert, H.. The exact approximation order in the central limit theorem for random U-statistics. (Preprint, 1982).Google Scholar
[2]Aerts, M. and Callaert, H.. The convergence rate of sequential fixed-width confidence intervals for regular functionals. (Preprint, 1982.)Google Scholar
[3]Ahmad, I.. On the Berry-Esséen theorem for random U-statistics. Ann. Statist. 8 (1980), 1395–1398.CrossRefGoogle Scholar
[4]Anscombe, F. J.. Large sample theory for sequential estimation. Proc. Cambridge Philos. Soc. 48 (1952), 600–607.CrossRefGoogle Scholar
[5]Chow, Y. S. and Robbins, H.. On the asymptotic theorem of fixed-width sequential confidence intervals for the mean. Ann. Math. Statist. 36 (1965), 457–462.CrossRefGoogle Scholar
[6]Csenki, A.. On the convergence rate of fixed-width sequential confidence intervals. Scand. Actuar. J. (1980), 107–111.CrossRefGoogle Scholar
[7]Csenki, A.. A theorem on the departure of randomly indexed U-statistics from normality with an application in fixed-width sequential interval estimation. Sankhyā Ser. A. 43 (1981), 84–99.Google Scholar
[8]Csörgoő, M. and Révész, P.. Strong Approximations in Probability and Statistics. (Academic Press-Akadémiai Kiadó, 1981).Google Scholar
[9]Ghosh, M. and DasGupta, R.. Berry-Esséen theorems for U-statistics in the non i.i.d. In Trans. Colloquium on Nonparametric Stat. Inference. Coll. Math. Soc. J. Bolyai No. 32 (North-Holland, 1980), 293–314.Google Scholar
[10]Ghosh, M. and Mukhopadhyay, N.. Sequential point estimation of the mean when the distribution is unspecified. Comm. Statist. A. 8 (1979), 637–652.CrossRefGoogle Scholar
[11]Heyde, C. C.. Invariance principles in statistics. Int. Statist. Rev. 9 (1981), 143–152.CrossRefGoogle Scholar
[12]Horváth, L.. Strong approximation of renewal processes. Stochastic. Processes Appl. 18 (1984), 127–138.CrossRefGoogle Scholar
[13]Horváth, L.. Strong approximations of renewal processes and their applications. Acta Sci. Math. Hung. (To appear.)Google Scholar
[14]Loynes, R. M.. On the weak convergence of U-statistics processes and of the empirical process. Math. Proc. Cambridge Philos. Soc. 83 (1978), 269–272.CrossRefGoogle Scholar
[15]Miller, R. G. Jr and Sen, P. K.. Weak convergence of U-statistics and von Mises' differentiable statistical functions. Ann. Math. Statist. 43 (1972), 31–41.CrossRefGoogle Scholar
[16]Robbins, H.. Sequential estimation of the mean of a normal population. In Probability and Statistics – H. Cramér Volume (Almquist and Wiksell, 1959), 235–245.Google Scholar
[17]Sen, P. K. and Ghosh, M.. Sequential point estimation of estimable parameters based on U-statistics. Sankhya Ser. A. 43 (1981), 331–344.Google Scholar
[18]Serfling, R. J.. Approximation Theorems of Mathematical Statistics (Wiley, 1980).CrossRefGoogle Scholar
[19]Sproule, R. N.. A sequential fixed-width confidence interval for the mean of a U-statistic. Ph.D. thesis, University of North Carolina at Chapel Hill (1969).Google Scholar
[20]Sproule, R. N.. Asymptotic properties of U-statistics. Trans. Amer. Math. Soc. 199 (1974), 55–64.Google Scholar
[21]Vervaat, W.. Functional central limit theorems for processes with positive drift and their inverses. Z. Wahrsch. Verw. Qebiete 23 (1972), 245–253.CrossRefGoogle Scholar