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1995 Fiscal Year Final Research Report Summary

Inference on random partition models.

Research Project

Project/Area Number 06680288
Research Category

Grant-in-Aid for General Scientific Research (C)

Allocation TypeSingle-year Grants
Research Field Statistical science
Research InstitutionKeio University,

Principal Investigator

SIBUYA Masaaki  Department of Mathematics, Keio University.Professor., 理工学部・数理科学科, 教授 (20146723)

Co-Investigator(Kenkyū-buntansha) TAKEUCHI Juichiro  Dept.of Administration Engineering, Keio University.Lecturer., 理工学部・管理工学科, 専任講師 (80051697)
Project Period (FY) 1994 – 1995
Keywordsurn model / GEM distribution / Ewens sampling formula / record breaking / independence test / UMP test
Research Abstract

A.Regard a random partition of a finite set as a process to throw balls into urns, and note that the distinguishability or undistinguishability of balls and urns are essential in the model. There are four cases by the distinction. If the random partition is consistent in these cases and for any number of balls, there are some regularity in probability laws :
(1) The probability disribution depends on the number of balls in urns, but not on what balls.
(2) Along with the sequence number of urns, the number of balls has a Markovian property.
(3) If the number of balls is changed, the distribution of less balls is a marginal distribution of the one of more balls.
These three properties are essential in fundamental random partitions. Some of the properties are enough to characterize the four kinds of families of partitions with a single positive parameter. (Japan J.Industr.Appl.Math.)
B.In the ecoligical study of the variety of species, frequently used is the "residual allocation model", which … More generate the GEM distribution on the standard simplex of of the infinite dimension. The random partition of undistinguishable balls is the discrete version of the model.
As the number of balls increses to infinity, the ratio of balls in urns approaches to the GEM distribution. Based on the relationship, we characterize the GEM distribution in terms of the urn model and vice versa. (Statistics and Probability Letters).
C.The four kinds of families of random partitions with a parameter alpha>0 has the common minimal sufficient statistic, the number of urns occupied by balls. The number has the power series type distribution (the discrete exponential-type), and it is, when alpha=1, the distribution of the number of upper new records of any i.i.d. sequence of continuous random variables.
Hence, the new-record-number test of the independence is the Uniformly Most Powerful nonparametric test against alpha>1. Its power is compared with kendall's gamma test, and it is discussed when such an alternative will appear. Tokyo monthly average temperature, in August in the past 45 years, is significant by our test, but not so by gamma test.(Japan.J.Applied Statistics.). Less

  • Research Products

    (7 results)

All Other

All Publications (7 results)

  • [Publications] M. Sibuya and H. Yamato: "Characterization of some random partitions" Japan J. Industrial and Applied Math.12. 237-263 (1995)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M. Sibuya and H. Yamato: "Ordered and unordered randum partition of an integer and the GEM distribution" Statistics and Probability Letters. 25. 173-183 (1995)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 飯山由利子,西村和夫,渋谷政昭: "記録数検定の検出力" 応用統計学. 24. 13-26 (1995)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M. Sibuya: "The shape of a probability density function and its hazard function" Jewell. N. P. et al. (eds.) Lifetime Data,Kluwer. 307-314 (1995)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Sibuya, M.and Yamato, H.: "Characterization of some random partitions" Japan J.Industr.APPI.Math.12. 237-263 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Sibuya, M.and Yamato, H.: "Ordered and unordered random partition of an integer and the GEM distribution" Statist.Probab.Letters. 25. 177-183 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Iiyama, Y.Nishimura, K.and Sibuya, M.: "Power of recordbreaking test" Jaoanese J.Applied Statistics. 24. 13-26 (1995)

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

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Published: 1997-03-04  

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