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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.


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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

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