2005 Fiscal Year Final Research Report Summary
Study on functional s of distributions and convex combinations of U-statistics
| Project/Area Number |
15540130
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kagoshima University |
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Principal Investigator |
YAMATO Hajime Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (90041227)
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| 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)
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| Project Period (FY) |
2003 – 2005
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| Keywords | U-statistic / V -statistic / convex combination / estimable parameter |
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| 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.
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