Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T21:42:08.837Z Has data issue: false hasContentIssue false
  • English
  • Français

Inequalities for multinomial allocations with application to DNA fingerprinting

Part of: Combinatorial probability

Published online by Cambridge University Press:  14 July 2016

Sergei Grishechkin
Sergei Grishechkin*
Affiliation:
Moscow State University
*
∗Postal address: Warshavskoje shosse, d.88, kv.43, Moscow 113556, Russia. Email address: serge@star.net
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider an allocation of n balls into N cells according to probabilities pi. Assuming that the balls are allocated successively, denote by φ(n,N) the number of such balls which go into an already occupied cell. If n = 2 the probability that two balls will occupy the same cell is equal to the so-called match probability MP = p21 + … + p2N. An upper estimate for the probability ℙ(φ(n,N) ≤ m) which depends only on n and MP is derived. Such inequalities are important for estimation of the reliability of DNA fingerprinting, a new method of crime investigation which is currently much debated.

Keywords

Random allocationsPoisson approximation

MSC classification

Primary: 60C05: Combinatorial probability
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 3 , September 1998 , pp. 707 - 717
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work supported by the Russian Foundation for Basic Research, grant 95–0099 and the Russian Human Scientific Foundation, grant 96-03-04379.

References

Arratia, R., Goldstein, L., and Gordon, L. (1990). Poisson approximation and the Chen–Stein method. Statist. Sci. 5, 403–434.Google Scholar
Chistyakov, V., Kolchin, V., and Sevastyanov, B. (1978). Random Allocations. Winston & Sons, Washington, DC.Google Scholar
Cox, D. R., and Hinkley, D. V. (1974). Theoretical Statistics. Chapman & Hall, London.Google Scholar
Herrin, G. (1993). Probability of matching RFLP patterns from unrelated individuals. Am. J. Hum. Genet. 52, 491–497.Google Scholar
Mihailov, V. G., and Zubkov, A. M. (1978). An estimate of the accuracy of the Poisson approximation in the problem of the distribution of particles into cells. Teor. Verojatnost. i Primen. 23, 819–824. (In Russian). English transl.: Theor. Prob. Appl. 23, 789–794.Google Scholar
National Research Council (1992). DNA Technology in Forensic Science. National Academy Press, Washington, DC.Google Scholar
National Research Council (1996). The Evaluation of Forensic DNA Evidence. National Academy Press, Washington, DC.Google Scholar
Riordan, J. (1958). An Introduction to Combinatorial Analysis. Wiley, New York.Google Scholar
Risch, N. J., and Delvin, B. (1992). On the probability of matching DNA fingerprints. Science 255, 717–720.Google Scholar
Roeder, K. (1994). DNA fingerprinting: a review of the controversy. Statist. Sci. 9, 222–278.Google Scholar
Sudbury, A. (1994). Comment. Statist. Sci. 9, 262–263.CrossRefGoogle Scholar
Weir, B. S. (1995). A bibliography for the use of DNA in human identification. In Human Identification: The Use of DNA Markers. Ed. Weir, B.S. Kluwer, Dordrecht, pp. 179–213.CrossRefGoogle Scholar