| Project/Area Number |
13640124
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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) |
ATSUMI Tsuyosi Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (20041238)
KONDO Masao Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (70117505)
INADA Kouichi Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (20018899)
NOMACHI Toshifumi Miyakonojyou College of Technology, Associate Prof, 助教授 (70228352)
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| Project Period (FY) |
2001 – 2002
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| Project Status |
Completed (Fiscal Year 2002)
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| Budget Amount *help |
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2002: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2001: ¥1,700,000 (Direct Cost: ¥1,700,000)
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| Keywords | U-statistics / V-statistics / Asymptotic efficiency / Invariance principle / Asymptotic distribution / LB-統計量 / 高次有効性 |
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| Research Abstract |
As estimators of an estimable parameter, U-statistics and V-statistics are well-known and have been discussed. We get the new statistic which are described by a linear combination of U-statistics and include these statistics. By introducing a new weight function we can get a new statistic. Our results are as follows: (1) By a new weight, we get a new S-statistic. For a non-degenerate kernel, V-statistic and S-statistic have no difference in the mean of second order efficiency. So we compare these two statistics by 4-th order efficiency. (2) As the sample size increases, the behavior of the linear combination of U-statistics can be described by a stochastic process on the interval [0,1]. We show the invariance principle that this process approaches to Brown motion. (3) The rate of convergence of the linear combination of U-statistics are evaluated with respect to the almost sure convergence. (4) We get the Edgeworth expansion of the linear combination of U-statistics. The above results are obtained for non-degenerate kernel. For degenerate kernel, the asymptotic distribution is complicated. Our results for degenerate kernel are as follows: (5) We get the asymptotic distribution of the linear combination of U-statistics. This result yields the new evaluation of the asymptotic distribution of the V-statistic, which is different from the previous fact. (6) We show the invariance principle of the linear combination of U-statistics. (7) We evaluate the rate of convergence that the linear combination of U-statistic approaches to the asymptotic distribution for degenerate kernel.
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