Metrika

, Volume 47, Issue 1, pp 47–69

Limit theorems for multivariate discrete distributions

Article

Abstract

According to the usual law of small numbers a multivariate Poisson distribution is derived by defining an appropriate model for multivariate Binomial distributions and examining their behaviour for large numbers of trials and small probabilities of marginal and simultaneous successes. The weak limit law is a generalization of Poisson's distribution to larger finite dimensions with arbitrary dependence structure. Compounding this multivariate Poisson distribution by a Gamma distribution results in a multivariate Pascal distribution which is again asymptotically multivariate Poisson. These Pascal distributions contain a class of multivariate geometric distributions. Finally the bivariate Binomial distribution is shown to be the limit law of appropriate bivariate hypergeometric distributions. Proving the limit theorems mentioned here as well as understanding the corresponding limit distributions becomes feasible by using probability generating functions.

Key words

Binomial distribution Poisson distribution Pascal distribution multivariate distributions dependence structure weak convergence probability generating functions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aitken AC (1936) A further note on multivariate selection. Proceedings of the Edinburgh Mathematical Society (Series 2) 5: 37–40MATHGoogle Scholar
  2. 2.
    Aki S (1985) Discrete distributions of order k on a binary sequence. Annals of the Institute of Statistical Mathematics 37: 205–224MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Beutler FJ (1983) A note on multivariate Poisson flows on stochastic processes. Advances in Applied Probability 15: 219–220.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bickel RJ, Freedman DJ (1981) Some asymptotic theory for the bootstrap. Annals of Statistics 9: 1196–1217.MATHMathSciNetGoogle Scholar
  5. 5.
    Block HW Savitz TH, Shaked M (1982) Some concepts of negative dependence. Annals of Probability 10: 765–772MATHMathSciNetGoogle Scholar
  6. 6.
    Cacoullos T, Papageorgiou H (1983) Characterizations of discrete distributions by a conditional distribution and a regression function. Annals of the Institute of Statistical Mathematics 35: 95–104.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Campbell JT (1934) The Poisson correlation function. Proceedings of the Edinburgh Mathematical Society (Series 2) 4: 18–26MATHGoogle Scholar
  8. 8.
    Dawass M, Teicher H (1956) On infinitely divisible random vectors. Annals of Mathematical Statistics 27: 461–470Google Scholar
  9. 9.
    Ehm W (1991) Binomial approximation to the Poisson binomial distribution. Stat. Probability Letters 11: 7–16MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Famoye F, Consul PC (1995) Bivariate generalized Poisson distribution with some applications. Metrika 42: 127–139MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gordon FS, Gordon SP (1975) Transcendental functions of a vector variable and a characterization of a multivariate Poisson distribution. Statistical Distributions in Scientific Work 3: 163–172.Google Scholar
  12. 12.
    Gourieroux C, Monfort A (1979) On the characterization of a joint probability distribution by conditional distributions. Journal of Econometrics 10: 115–118CrossRefMathSciNetGoogle Scholar
  13. 13.
    Griffiths RC, Milne RK (1978) A class of bivariate Poisson processes. Journal of Multivariate Analysis 8: 380–395MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Griffiths RC, Milne RK, Wood R (1979) Aspects of correlation in bivariate Poisson distributions and processes. Australian Journal of Statistics 21: 238–255MATHMathSciNetGoogle Scholar
  15. 15.
    Hamadan MA (1973) A stochastic derivation of the bivariate Poisson distribution. South African Statistical Journal 7: 69–71MathSciNetGoogle Scholar
  16. 16.
    Hamadan MA, Al-Bayyati HA (1971) Canonical expansion of the compound correlated bivariate Poisson distribution. Journal of the American Statistical Association 66: 390–393CrossRefGoogle Scholar
  17. 17.
    Hawkes AG (1972) A bivariate exponential distribution with applications to reliability. Journal of the Royal Statistical Society, SeriesB, 34: 129–121MATHGoogle Scholar
  18. 18.
    Holgate P (1964) Estimation for the bivariate Poisson distribution. Biometrika 51: 241–245MATHMathSciNetGoogle Scholar
  19. 19.
    Jamardan KG (1973) Chance mechanisms for multivariate hypergeometric models. Sankhya A 35: 465–478Google Scholar
  20. 20.
    Jamardan KG (1975) Certain inference problems for multivariate hypergeometric distributions. Communications in Statistics 4: 375–388CrossRefGoogle Scholar
  21. 21.
    Jamardan KG (1976) Certain estimation problems for multivariate hypergeometric models. Annals of the Institute of Statistical Mathematics 28: 429–444CrossRefMathSciNetGoogle Scholar
  22. 22.
    Jensen DR (1971) A note on Positive dependence and the structure of bivariate distributions. SIAM Journal of Applied Mathematics 20: 749–752MATHCrossRefGoogle Scholar
  23. 23.
    Jogedo K (1968) Characterizations of independence in certain families of bivariate and multivariate distributions. Annals of Mathematical Statistics 39: 433–441MathSciNetGoogle Scholar
  24. 24.
    Jogedo K (1975) Dependence concepts and probability inequalities. Statistical Distributions in Scientific Work 1: 271–279Google Scholar
  25. 25.
    Jogedo K, Patil GP (1975) Probability inequalities for certain multivariate discrete distributions. Sankhya B, 37: 158–164.Google Scholar
  26. 26.
    Johnson N, Kotz S (1969) Distributions in statistics I: Discrete distributions. Houghton Mifflin BostonMATHGoogle Scholar
  27. 27.
    Johnson N, Kotz S (1972) Distributions in statistics IV: Continuous multivariate distributions. John Wiley, New York.MATHGoogle Scholar
  28. 28.
    Khatri CG (1971) On multivariate contagious distributions. Sankhya B, 33: 197–216MathSciNetGoogle Scholar
  29. 29.
    Kocherlakota S, Kocherlakota K (1993) Bivariate discrete distributions. Marcel Dekker, New YorkGoogle Scholar
  30. 30.
    Krishnamoorthy AS (1951) Multivariate binomial and Poisson distributions. Sankhya 11: 117–124MATHMathSciNetGoogle Scholar
  31. 21.
    Krummenauer F (1996) Grenzwertsätze für multivariate diskrete Verteilungen. Doctoral Thesis in Statistics, University of Dortmund, GermanyGoogle Scholar
  32. 32.
    Lukacs E (1979) Some multivariate statistical characterization theorems. Journal of Multivariate Analysis 9: 278–287MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Lukacs E, Beer S (1977) Characterization of the multivariate Poisson distribution. Journal of Multivariate Analysis 7: 1–12MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Marshall AW, Olkin I (1985) A family of bivariate distributions generated by the bivariate Bernoulli distribution. Journal of the American Statistical Association 80: 332–338.MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Mitchell CR, Paulson AS (1981) A new bivariate negative binomial distribution. Naval Research Logistics Quaterly 28: 359–374MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Nelson JF (1985) Multivariate Gamma-Poisson models. Journal of the American Statistical Association 8: 828–834CrossRefGoogle Scholar
  37. 37.
    Nevill AM, Kemp CD (1975) On characterizing the hypergeometric and multivariate hypergeometric distributions. Statistical Distributions in Scientific Work 3: 353–358Google Scholar
  38. 38.
    Panaretos J (1980) A characterization of a general class of multivariate discrete distributions. In: Gyeres (ed.) Analytic function methods in probability theory Colloquia Mathematica Societatis Janos Bolyai, 21, North Holland, Amsterdam, pp. 243–252Google Scholar
  39. 39.
    Panaretos J (1983) An elementary characterization of the multinomial and the multivariate hypergeometric distribution. In: Kalashnikov VV, Zolotarev, VM (eds) Statistical problems for stochastic models. Springer-Verlag, New York, pp. 156–164.CrossRefGoogle Scholar
  40. 40.
    Patil GP (1964) On certain compound Poisson and compound binomial distributions. Sankhya A 26: 293–294MATHMathSciNetGoogle Scholar
  41. 41.
    Sibuya M (1983) Generalized hypergeometric distributions. Encyclopedia of Statistical Sciences 3: 330–334Google Scholar
  42. 42.
    Steyn HS (1951) On discrete multivariate probability functions of hypergeometric type. Koninklijke Nederlandse Akademie Wetenschappen Proceedings (Series A) 54: 23–30.MathSciNetGoogle Scholar
  43. 43.
    Steyn HS (1955) On discrete multivariate probability functions of hypergeometric type. Koninklijke Nederlandse Akademie Wetenschappen Proceedings (Series A) 58: 588–595MathSciNetMATHGoogle Scholar
  44. 44.
    Takeuchi K, Takemura A (1987) On sums of 0–1 random variables II: Multivariate case. Annals of the Institute of Statistical Mathematics 39, Part A: 307–324MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Talwalker S (1970) A characterization of the double Poisson distribution. Sankhya A, 32: 265–270MATHMathSciNetGoogle Scholar
  46. 46.
    Teicher H (1954) On the multivariate Poisson distribution. Scandinavisk Aktuarietidskrift 37: 1–9MathSciNetGoogle Scholar
  47. 47.
    Wang YH (1989) A multivariate extension of Poisson's theorem. Canadian Journal of Statistics 17: 241–245MATHCrossRefGoogle Scholar
  48. 48.
    Wicksell SD (1923) Contributions to the analytical theory of sampling. Arkiv for Matematik, Astronomi Och Fysik 17: 1–46Google Scholar
  49. 49.
    Zolotarev VM (1966) A multidimensional analogue of the Berry/Esséen inequality for sets with bounded diameter. SIAM Theory of Probability and Its Applications 11: 447–454.CrossRefGoogle Scholar

Copyright information

© Physica-Verlag 1998

Authors and Affiliations

  1. 1.Institut für Medizinische Statistik und DokumentationMainzGermany

Personalised recommendations