1995 Fiscal Year Final Research Report Summary
Inference on random partition models.
| Project/Area Number |
06680288
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
Statistical science
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| Research Institution | Keio University, |
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Principal Investigator |
SIBUYA Masaaki Department of Mathematics, Keio University.Professor., 理工学部・数理科学科, 教授 (20146723)
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| 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)
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| Project Period (FY) |
1994 – 1995
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| Keywords | urn model / GEM distribution / Ewens sampling formula / record breaking / independence test / UMP test |
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| 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
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