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A Class of Statistics with Asymptotically Normal Distribution

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Breakthroughs in Statistics

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Abstract

Let X 1 …, X n be n independent random vectors, X v = , and Φ(x 1 …, x m ) a function of m(≤n) vectors . A statistic of the form , where the sum ∑″ is extended over all permutations (α1 …, α m ) of different integers, 1 α≤ (αi≤ n, is called a U-statistic. If X 1, …, X n have the same (cumulative) distribution function (d.f.) F(x), U is an unbiased estimate of the population characteristic θ(F) = f … f Φ(x 1,…, x m ) dF(x 1) … dF(x m ). θ(F) is called a regular functional of the d.f. F(x). Certain optimal properties of U-statistics as unbiased estimates of regular functionals have been established by Halmos [9] (cf. Section 4)

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Author information

Authors and Affiliations

  1. Institute of Statistics University of North Carolina, USA

    Wassily Hoeffding

Authors
  1. Wassily Hoeffding
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Editor information

Editors and Affiliations

  1. College of Business and Management, University of Maryland at College Park, College Park, MD, 20742, USA

    Samuel Kotz

  2. Department of Statistics Phillips Hall, The University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599, USA

    Norman L. Johnson

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© 1992 Springer Science+Business Media New York

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Hoeffding, W. (1992). A Class of Statistics with Asymptotically Normal Distribution. In: Kotz, S., Johnson, N.L. (eds) Breakthroughs in Statistics. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0919-5_20

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  • DOI: https://doi.org/10.1007/978-1-4612-0919-5_20

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-94037-3

  • Online ISBN: 978-1-4612-0919-5

  • eBook Packages: Springer Book Archive

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