Asymptotic properties of $U$-statistics
HTML articles powered by AMS MathViewer
- by Raymond N. Sproule PDF
- Trans. Amer. Math. Soc. 199 (1974), 55-64 Request permission
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
- F. J. Anscombe, Large-sample theory of sequential estimation, Proc. Cambridge Philos. Soc. 48 (1952), 600–607. MR 51486, DOI 10.1017/s0305004100076386
- Robert H. Berk, Limiting behavior of posterior distributions when the model is incorrect, Ann. Math. Statist. 37 (1966), 51–58; correction, ibid. 745–746. MR 189176, DOI 10.1214/aoms/1177699477 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.
- Wassily Hoeffding, A class of statistics with asymptotically normal distribution, Ann. Math. Statistics 19 (1948), 293–325. MR 26294, DOI 10.1214/aoms/1177730196 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.
- Maurice G. Kendall and Alan Stuart, The advanced theory of statistics. Vol. 1. Distribution theory, Hafner Publishing Co., New York, 1958. MR 0124940
- 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 300321, DOI 10.1214/aoms/1177692698
- A. Rényi, On mixing sequences of sets, Acta Math. Acad. Sci. Hungar. 9 (1958), 215–228. MR 98161, DOI 10.1007/BF02023873
- 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 115204, DOI 10.1007/BF02020627 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. F. Wilcoxon, Individual comparison by ranking methods, Biometrics Bull. 1 (1945), 80-83.
Additional Information
- © Copyright 1974 American Mathematical Society
- 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