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2005 Fiscal Year Final Research Report Summary

Study on functional s of distributions and convex combinations of U-statistics

Research Project

Project/Area Number 15540130
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKagoshima University

Principal Investigator

YAMATO Hajime  Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (90041227)

Co-Investigator(Kenkyū-buntansha) NADA Koiichi  Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (20018899)
KONDO Masao  Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (70117505)
Project Period (FY) 2003 – 2005
KeywordsU-statistic / V -statistic / convex combination / estimable parameter
Research Abstract

As statistics related with a functional of distribution, V-statistic of von Mises (1947) and U-statistic of Hoeffding (1948) are well-known and have been studied by many researchers. Yamato (1977) introduced a new statistic, LB-statistic. Toda and Yamato (2001) introduced a linear combination of U-statistics (a convex combination of U-statistic), which include these three statistics.
We have studied the properties of this convex combination of U-statistics.
We evaluated large deviations of the convex combination of U-statistics, which is useful to evaluate the tail probability. In case where the kernel is degenerate, we obtained a functional limit theorem (invariance principle) of a convex combination of U-statistics.
We also obtained its Edgeworth expansion. At first we got it for the standardized statistic by the order-1 of the sample size and furthermore got it for the studentized statistic by the order -1/2 of the sample size. We also got the Edgeworth expansion for the studentized statistic by the order-1 of the sample size.
Next, we introduced a convex combination of two-sample U-statistics. This statistic has the asymptotic normality as same as the two-sample U-statistic. To see the difference between these two statistics, we got the Edgeworth expansion of convex combination of two-sample U-statistics with the order -1/2 of the sample size.

  • Research Products

    (10 results)

All 2006 2005 2004 2003

All Journal Article (10 results)

  • [Journal Article] A convex combination of two-sample U-statistics2006

    • Author(s)
      Koichiro Toda
    • Journal Title

      J.Japan Statistical Society (掲載決定済み)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A convex combination of two-sample U-statistics2006

    • Author(s)
      Koichiro Toda
    • Journal Title

      J. Japan Statistical Society (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Asymptotic properties of a linear combination of U-statistics with Degenerate kernel2005

    • Author(s)
      Hajime Yamato
    • Journal Title

      J.Nonparametric Statistics 17・2

      Pages: 187-199

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Asymptotic properties of a linear combination of U-statistics with degenerate kernel2005

    • Author(s)
      Hajime Yamato
    • Journal Title

      J Nonparametric Statistics 17-2

      Pages: 187-199

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] An Edgeworth expansion of a convex combination of U-statistics Based on studentization2004

    • Author(s)
      Hajime Yamato
    • Journal Title

      Bulletin of Informatics and Cybernetics 36・1

      Pages: 105-130

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] An Edgeworth expansion of a convex combination of U-statistics based on studentization.2004

    • Author(s)
      Hajime Yamato
    • Journal Title

      Bulletin of Informatics and Cybernetics 36-1

      Pages: 105-130

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Large deviation for linear combination of U-statistics2003

    • Author(s)
      Hajime Yamato
    • Journal Title

      Scientiae Mathematicae Japonicae 57・2

      Pages: 207-215

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Edgeworth expansions of some statistics including the LB-statistics and V-statistic2003

    • Author(s)
      Hajime Yamato
    • Journal Title

      J.Japan Statistical Society 33・1

      Pages: 77-94

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Large deviations for a linear combination of U-statistics2003

    • Author(s)
      Hajime Yamato
    • Journal Title

      Scientiae Mathematicae Japonicae 57-2

      Pages: 207-215

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Edgeworth expansions of some statistics including the LB-statistic and V -statistic2003

    • Author(s)
      Hajime Yamato
    • Journal Title

      J. Japan Statistical Society 33-1

      Pages: 77-94

    • Description
      「研究成果報告書概要(欧文)」より

URL: 

Published: 2007-12-13  

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