Asymptotic properties of -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 be a fixed positive integer. A -statistic is an average of a symmetric measurable function of arguments over a random sample of size . 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 -statistic. In addition, the asymptotic normality of , where 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,
-statistics,
large sample properties,
law of large numbers,
martingales,
central limit theorem,
the sample mean

Article copyright:
© Copyright 1974
American Mathematical Society