Advertisement

Measures on Partitions

Chapter
Part of the SpringerBriefs in Statistics book series (BRIEFSSTATIST)

Abstract

After brief introduction of the multiplicative measure, defined as a family of measures on integer partitions, which include typical combinatorial structures, this chapter introduces the exponential structure, which plays important roles in statistical inference. It then introduces the Gibbs partition, a generalization of the exponential structure. The generalization is achieved by systematic use of partial Bell polynomials. Gibbs partitions characterize prior processes in Bayesian nonparametrics, and appear as statistical models of diversity in count data. The Ewens sampling formula and the Pitman partition are well-known examples of Gibbs partitions. Finally, this section discusses the asymptotic behaviors of extremes of the sizes of parts in Gibbs partitions. Some of the results are derived by simple analytic approaches.

Keywords

Analytic combinatorics Asymptotics Ewens sampling formula Exponential structure Extreme Gibbs partition Multiplicative measure Partial bell polynomial Pitman partition 

References

  1. 1.
    Logan, B.F., Shepp, L.A.: A variational problem for random Young tableaux. Adv. Math. 26, 206–222 (1977)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Vershik, A.M., Kerov, S.V.: Asymptotic behavior of the Plancherel measure of the symmetric group and the limit from of Young tableaux. Dokl. Akad. Nauk SSSR 233, 1024–1037 (1977)MathSciNetGoogle Scholar
  3. 3.
    Kerov, S.V.: Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis, vol. 219. American Mathematical Society, Providence (2003). Translations of Mathematical MonographGoogle Scholar
  4. 4.
    Borodin, A., Corwin, I.: Macdonald processes. Probab. Theor. Relat. Fields 158, 225–400 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Vershik, A.M.: Statistical mechanics of combinatorial partitions, and their limit configurations. Funct. Anal. Appl. 30, 90–105 (1996)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Green, M.B., Schwarz, J.H., Witten, E.: Superstring Theory, vol. 2. Cambridge University Press, New York (1987)MATHGoogle Scholar
  7. 7.
    Nekrasov, N.A., Okounkov, A.: Seiberg-Witten theory and random partitions. In: Etingof, P., Retakh, V., Singer, I.M. (eds.) The Unity of Mathematics. Progress in Mathematics, vol. 244, pp. 525–596. Birkhäuser, Boston (2006)CrossRefGoogle Scholar
  8. 8.
    Takemura, A., et al.: Special issue: statistical theory of statistical disclosure control problem. Proc. Inst. Stat. Math. 51, 181–388 (2003)Google Scholar
  9. 9.
    Arratia, R., Barbour, A.D., Tavaré, S.: Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society, Zürich (2003). EMS Monographs in MathGoogle Scholar
  10. 10.
    Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, New York (1999)CrossRefGoogle Scholar
  11. 11.
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  12. 12.
    Flajolet, P., Soria, M.: Gaussian limiting distributions for the number of components in combinatorial structures. J. Combin. Theor. Ser. A 53, 165–182 (1990)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Corless, R., Gonnet, G., Hare, D., Jeffrey, D., Knuth, D.: On the Lambert W function. In: Advances in Computational Mathematics, vol. 5, pp. 329–359. Springer, Heidelberg (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Katz, L.: Probability of indecomposability of a random mapping function. Ann. Math. Stat. 26, 512–517 (1955)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ewens, W.J.: The sampling theory of selectively neutral alleles. Theor. Popul. Biol. 3, 87–112 (1972)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Antoniak, C.: Mixture of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Statist. 2, 1152–1174 (1974)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Sibuya, M.: A random-clustering process. Ann. Inst. Stat. Math. 45, 459–465 (1993)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tavaré, S., Ewens, W.J.: The Ewens sampling formula. In: Johnson, N.L., Kotz, S., Balakrishnan, N. (eds.) Multivariate Discrete Distributions. Wiley, New York (1997)Google Scholar
  19. 19.
    Crane, H.: The ubiquitous Ewens sampling formula. Stat. Sci. 31, 1–19 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Bell, E.T.: Exponential polynomials. Ann. Math. 35, 258–277 (1934)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Pitman, J.: Combinatorial stochastic processes. Ecole d’Été de Probabilités de Saint Flour, Lecture Notes in Mathematics, vol. 1875. Springer, Heidelberg (2006)Google Scholar
  22. 22.
    Comtet, L.: Advanced Combinatorics. Ridel, Dordrecht (1974)CrossRefGoogle Scholar
  23. 23.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, New York (1995)Google Scholar
  24. 24.
    Okano, T., Okuto, Y., Shimizu, A., Niikura, Y., Hashimoto, Y., Yamada, H. The formula of Faà di Bruno and its applications I. Annual Review 2000, Institute of National Sciences, Nagoya City University, pp. 35–44 (2000). in JapaneseGoogle Scholar
  25. 25.
    Kolchin, V.F.: A problem on the distribution of particles among cells, and cycles of random permutations. Teor. Veroyatnost. i Primenen. 16, 67–82 (1971)Google Scholar
  26. 26.
    Kerov, S.V: Coherent random allocations, and the Ewens-Pitman formula. Zap. Nauchn. Semi. POMI 325, 127–145 (1995); English translation: J. Math. Sci. 138, 5699–5710 (2006)Google Scholar
  27. 27.
    Hoshino, N.: Random partitioning over a sparse contingency table. Ann. Inst. Stat. Math. 64, 457–474 (2012)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Fisher, R.A., Corbet, A.S., Williams, C.B.: The relationship between the number of species and the number of individuals in a random sample of an animal population. J. Animal Ecol. 12, 42–58 (1943)CrossRefGoogle Scholar
  29. 29.
    Hoshino, N., Takemura, A.: Relationship between logarithmic series model and other superpopulation models useful for microdata discrosure risk assesment. J. Jpn. Stat. Soc. 28, 125–134 (1998)CrossRefGoogle Scholar
  30. 30.
    Bethlehem, J.G., Keller, W.J., Pannekoek, J.: Disclosure control of microdata. J. Am. Stat. Assoc. 85, 38–45 (1990)CrossRefGoogle Scholar
  31. 31.
    Hoshino, N.: On a limiting quasi-multinomial distribution. Discussion Paper CIRJE-F-361, Center for International Research on the Japanese Economy, Faculty of Economics, The University of Tokyo (2005)Google Scholar
  32. 32.
    Diaconis, P., Lam, A.: A probabilistic interpretation of the Macdonald polynomials. Ann. Probab. 40, 1861–1896 (2012)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Charalambides, C.A.: Combinatorial Methods in Discrete Distributions. Wiley, New Jersey (2005)CrossRefGoogle Scholar
  34. 34.
    Pitman, J.: Exchangeable and partially exchangeable random partitions. Probab. Theor. Relat. Fields 102, 145–158 (1995)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Hoshino, N.: A discrete multivariate distribution resulting from the law of small numbers. J. Appl. Probab. 43, 852–866 (2006)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Hoshino, N.: Engen’s extended negative binomial model revisited. Ann. Inst. Stat. Math. 57, 369–387 (2005)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Fisher, R.A.: Tests of significance in harmonic analysis. Proc. Roy. Soc. Lond. Ser. A 125, 54–59 (1929)CrossRefGoogle Scholar
  38. 38.
    Goncharov, V.L.: Some facts from combinatorics. Izv. Akad. Nauk SSSR, Ser. Mat. 8, 3–48 (1944)Google Scholar
  39. 39.
    Shepp, L.A., Lloyd, S.P.: Ordered cycle length in random permutation. Trans. Am. Math. Soc. 121, 340–357 (1966)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Tenenbaum, G.: Introduction to Analytic and Probabilistic Number Theory. Cambridge University Press, New York (1955)Google Scholar
  41. 41.
    Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New Jersey (1999)CrossRefGoogle Scholar
  42. 42.
    Panario, D., Richmond, B.: Smallest components in decomposable structures: exp-log class. Algorithmica 29, 205–226 (2010)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Arratia, R., Barbour, A.D., Tavaré, S.: Poisson process approximations for the Ewens sampling formula. Ann. Appl. Probab. 2, 519–535 (1992)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Feng, S.: The Poisson–Dirichlet Distributions and Related Topics. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  45. 45.
    Flajolet, P., Odlyzko, A.: Singularity analysis of generating functions. SIAM J. Discrete Math. 3, 216–240 (1990)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Hwang, H.-K.: Asymptotic expansions for Stirling’s number of the first kind. J. Combin. Theor. Ser. A 71, 343–351 (1995)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Mano, S.: Extreme sizes in the Gibbs-type random partitions. Ann. Inst. Stat. Math. 69, 1–37 (2017)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Mano, S.: Partition structure and the \(A\)-hypergeometric distribution associated with the rational normal curve. Electron. J. Stat. 11, 4452–4487 (2017)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Pitman, J.: Brownian motion, bridge, excursion and meander characterized by sampling at independent uniform times. Electron. J. Probab. 4, 33pp (1999)Google Scholar
  50. 50.
    Keener, R., Rothman, E., Starr, N.: Distribution of partitions. Ann. Stat. 15, 1466–1481 (1978)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Korwar, R.M., Hollander, M.: Contribution to the theory of Dirichlet process. Ann. Probab. 1, 705–711 (1973)CrossRefGoogle Scholar
  52. 52.
    Pollard, H.: The representation of \(e^{-x^\lambda }\) as a Laplace integral. Bull. Am. Math. Soc. 52, 908–910 (1946)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Watterson, G.A.: The sampling theory of selectively neutral alleles. Adv. Appl. Probab. 6, 463–488 (1974)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Yamato, H.: Edgeworth expansions for the number of distinct components associated with the Ewens sampling formula. J. Jpn. Stat. Soc. 43, 17–28 (2013)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Yamato, H.: Poisson approximations for sum of Bernoulli random variables and its application to Ewens sampling formula. J. Jpn. Stat. Soc. 47, 187–195 (2018)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Hansen, J.C.: A functional central limit theorem for the Ewens sampling formula. J. Appl. Probab. 27, 28–43 (1990)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Tsukuda, K.: Functional central limit theorems in \(L^2(0,1)\) for logarithmic combinatorial assemblies. Bernoulli 24, 1033–1052 (2018)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Feng, S.: Large deviations associated with Poisson–Dirichlet distribution and Ewens sampling formula. Ann. Appl. Probab. 17, 1570–1595 (2007)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Tsukuda, K.: Estimating the large mutation parameter of the Ewens sampling formula. J. Appl. Probab. 54, 42–54 (2017)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Tsukuda, K.: On Poisson approximations for the Ewens sampling formula when the mutation parameter grows with the sample size. arXiv: 1704.06768
  61. 61.
    Griffiths, R.C.: On the distribution of points in a Poisson process. J. Appl. Probab. 25, 336–345 (1988)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Pitman, J., Yor, M.: The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25, 855–899 (1997)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Dickman, K.: On the frequency of numbers containing prime factors of a certain relative magnitude. Ark. Mat., Astronomi och Fysik. 22, 1–14 (1930)Google Scholar
  64. 64.
    Handa, K.: The two-parameter Poisson–Dirichlet point process. Bernoulli 15, 1082–1116 (2009)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Griffiths, R.C.: On the distribution of allele frequencies in a diffusion model. Theor. Popul. Biol. 15, 140–158 (1979)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Buchstab, A.A.: An asymptotic estimation of a general number-theoretic function. Mat. Sb. 44, 1239–1246 (1937)Google Scholar
  67. 67.
    Arratia, R., Tavaré, S.: Limit theorems for combinatorial structures via discrete process approximations. Random Structures Algorithms 3, 321–345 (1992)MathSciNetCrossRefGoogle Scholar
  68. 68.
    Sibuya, M.: Generalized hypergeometric, digamma and trigamma distributions. Ann. Inst. Stat. Math. 31, 373–390 (1979)MathSciNetCrossRefGoogle Scholar
  69. 69.
    Devroye, L.: A triptych of discrete distributions related to the stable law. Stat. Probab. Lett. 18, 349–351 (1993)MathSciNetCrossRefGoogle Scholar
  70. 70.
    Karlin, S.: Central limit theorems for certain infinite urn schemes. J. Math. Mech. 17, 373–401 (1967)MathSciNetMATHGoogle Scholar
  71. 71.
    Rouault, A.: Lois de Zipf et sources markoviennes. Ann. Inst. H. Poincaré Sect. B 14, 169–188 (1978)MathSciNetMATHGoogle Scholar
  72. 72.
    Yamato, H., Sibuya, M.: Moments of some statistics of Pitman sampling formula. Bull. Inf. Cybernet. 32, 1–10 (2000)MathSciNetMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.The Institute of Statistical MathematicsTachikawaJapan

Personalised recommendations