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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Asymptotic properties of $ U$-statistics


Author: Raymond N. Sproule
Journal: Trans. Amer. Math. Soc. 199 (1974), 55-64
MSC: Primary 60F15
DOI: https://doi.org/10.1090/S0002-9947-1974-0350826-7
MathSciNet review: 0350826
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Abstract: Let $ r$ be a fixed positive integer. A $ U$-statistic $ {U_n}$ is an average of a symmetric measurable function of $ r$ arguments over a random sample of size $ n$. Such a statistic may be expressed as an average of independent and identically distributed random variables plus a remainder term. We develop a Kolmogorov-like inequality for this remainder term as well as examine some of its (a.s.) convergence properties. We then relate these properties to the $ U$-statistic. In addition, the asymptotic normality of $ {U_N}$, where $ N$ is a positive integer-valued random variable, is established under certain conditions.


References [Enhancements On Off] (What's this?)

  • [1] F. J. Anscombe, Large-sample theory of sequential estimation, Proc. Cambridge Philos. Soc. 48 (1952), 600–607. MR 0051486
  • [2] Robert H. Berk, Limiting behavior of posterior distributions when the model is incorrect, Ann. Math. Statist. 37 (1966), 51–58; correction, ibid 37 (1966), 745–746. MR 0189176
  • [3] C. Gini, Sulla misura delta concentrazione e della variabilita dei caratteri, Atti del R. Istituto Veneto di S. L. A. 73 (1913/14), part 2.
  • [4] Wassily Hoeffding, A class of statistics with asymptotically normal distribution, Ann. Math. Statistics 19 (1948), 293–325. MR 0026294
  • [5] W. Hoeffding, The strong law of large numbers for $ U$-statistics, Institute of Statistics Mimeo Series No. 302, University of North Carolina, Chapel Hill, N. C., 1961.
  • [6] Maurice G. Kendall and Alan Stuart, The advanced theory of statistics. Vol. 1. Distribution theory, Hafner Publishing Co., New York, 1958. MR 0124940
  • [7] R. G. Miller Jr. and Pranab Kumar Sen, Weak convergence of 𝑈-statistics and von Mises’ differentiable statistical functions, Ann. Math. Statist. 43 (1972), 31–41. MR 0300321, https://doi.org/10.1214/aoms/1177692698
  • [8] A. Rényi, On mixing sequences of sets, Acta Math. Acad. Sci. Hungar. 9 (1958), 215–228. MR 0098161, https://doi.org/10.1007/BF02023873
  • [9] A. Rényi, On the central limit theorem for the sum of a random number of independent random variables, Acta Math. Acad. Sci. Hungar. 11 (1960), 97–102 (unbound insert) (English, with Russian summary). MR 0115204, https://doi.org/10.1007/BF02020627
  • [10] R. N. Sproule, A sequential fixed-width confidence interval for the mean of a $ U$-statistic, Institute of Statistics Mimeo Series No. 636, University of North Carolina, Chapel Hill, N. C., 1969.
  • [11] F. Wilcoxon, Individual comparison by ranking methods, Biometrics Bull. 1 (1945), 80-83.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0350826-7
Keywords: Nonparametric testing, almost sure convergence, asymptotic normality, Kolmogorov inequality, generalization of sample mean, $ U$-statistics, large sample properties, law of large numbers, martingales, central limit theorem, the sample mean
Article copyright: © Copyright 1974 American Mathematical Society