Advertisement

Poisson Approximation

  • Kenneth Lange
Part of the Statistics for Biology and Health book series (SBH)

Abstract

In the past few years, mathematicians have developed a powerful technique known as the Chen-Stein method [2, 4] for approximating the distribution of a sum of weakly dependent Bernoulli random variables. In contrast to many asymptotic methods, this approximation carries with it explicit error bounds. Let X α be a Bernoulli random variable with success probability p α where a ranges over some finite index set I. It is natural to speculate that the sum S = Σα∈1 X α is approximately Poisson with mean λ = Σα∈I p α. The Chen-Stein method estimates the error in this approximation using the total variation distance between two integer-valued random variables Y and Z.

Keywords

Somatic Cell Hybrid White Ball Bernoulli Random Variable Poisson Approximation Total Variation Distance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Arratia R, Goldstein L, Gordon L (1989) Two moments suffice for Poisson approximations: the Chen-Stein method. Ann Prob 17:9–25zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Arratia R, Goldstein L, Gordon L (1990) Poisson approximation and the Chen-Stein method. Stat Sci 5:403–434.zbMATHMathSciNetGoogle Scholar
  3. [3]
    Arratia R, Gordon L, Waterman MS (1986) An extreme value theory for sequence matching. Ann Stat 14:971–993zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Barbour AD, Holst L, Janson S (1992) Poisson Approximation. Oxford University Press, OxfordzbMATHGoogle Scholar
  5. [5]
    D’Eustachio P, Ruddle FH (1983) Somatic cell genetics and gene families. Science 220:919–924CrossRefGoogle Scholar
  6. [6]
    Flatto L, Konheim AG (1962) The random division of an interval and the random covering of a circle. SIAM Review 4:211–222zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Glaz J (1992) Extreme order statistics for a sequence of dependent random variables. Stochastic Inequalities, Shaked M, Tong YL, editors, IMS Lecture Notes — Monograph Series, Vol 22, Hay ward, CA, pp 100–115Google Scholar
  8. [8]
    Goradia TM (1991) Stochastic Models for Human Gene Mapping. Ph.D. Thesis, Division of Applied Sciences, Harvard UniversityGoogle Scholar
  9. [9]
    Goradia TM, Lange K (1988) Applications of coding theory to the design of somatic cell hybrid panels. Math Biosciences 91:201–219zbMATHCrossRefGoogle Scholar
  10. [10]
    Hille E (1959) Analytic Function Theory, Vol 1. Blaisdell, New YorkGoogle Scholar
  11. [11]
    Karlin S, Macken C (1991) Some statistical problems in the assessment of inhomogeneities of DNA sequence data. J Amer Stat Assoc 86:27–35CrossRefGoogle Scholar
  12. [12]
    Lange K, Boehnke M (1982) How many polymorphic genes will it take to span the human genome? Amer J Hum Genet 34:842–845Google Scholar
  13. [13]
    Lindvall T (1992) Lectures on the Coupling Method. Wiley, New YorkzbMATHGoogle Scholar
  14. [14]
    Roos M (1993) Compound Poisson approximations for the numbers of extreme spacings. Adv Appl Prob 25:847–874zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Rushton AR (1976) Quantitative analysis of human chromosome segregation in man-mouse somatic cell hybrids. Cytogenetics Cell Genet 17:243–253CrossRefGoogle Scholar
  16. [16]
    Schilling MF (1990) The longest run of heads. The College Mathematics Journal 21:196–207zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Solomon H (1978) Geometric Probability. SIAM, PhiladelphiazbMATHCrossRefGoogle Scholar
  18. [18]
    Waterman MS (1989) Sequence alignments. Mathematical Methods for DNA Sequences, CRC Press, Boca Raton, FL, pp 53–92Google Scholar
  19. [19]
    Waterman MS (1995) Introduction to Computational Biology: Maps, Sequences, and Genomes. Chapman and Hall, LondonzbMATHGoogle Scholar
  20. [20]
    Weiss M, Green H (1967) Human-mouse hybrid cell lines containing partial complements of human chromosomes and functioning human genes. Proc Natl Acad Sci 58:1104–1111CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Kenneth Lange
    • 1
  1. 1.Department of Biostatistics and MathematicsUniversity of MichiganAnn ArborUSA

Personalised recommendations