Coupon collector’s problems with statistical applications to rankings

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Abstract

Some new exact distributions on coupon collector’s waiting time problems are given based on a generalized Pólya urn sampling. In particular, usual Pólya urn sampling generates an exchangeable random sequence. In this case, an alternative derivation of the distribution is also obtained from de Finetti’s theorem. In coupon collector’s waiting time problems with \(m\) kinds of coupons, the observed order of \(m\) kinds of coupons corresponds to a permutation of \(m\) letters uniquely. Using the property of coupon collector’s problems, a statistical model on the permutation group of \(m\) letters is proposed for analyzing ranked data. In the model, as the parameters mean the proportion of the \(m\) kinds of coupons, the observed ranking can be intuitively understood. Some examples of statistical inference are also given.

Keywords

Generalized Pólya urn Dirichlet distribution Exchangeability Likelihood ratio test Permutation 

References

  1. Charalambides, Ch A. (2002). Enumerative combinatorics. Boca Raton: Chapman& Hall/CRC.MATHGoogle Scholar
  2. Charalambides, Ch A. (2005). Combinatorial methods in discrete distributions. New York: Wiley.MATHCrossRefGoogle Scholar
  3. Diaconis, P. (1988). Group representations in probability and statistics. Lecture notes—Monograph Series 11, IMS.Google Scholar
  4. Graham, R. L., Knuth, D. E., Patashnik, O. (1989). Concrete mathematics. Massachusetts: Addison-Wesley Publishing Company.Google Scholar
  5. Hall, P., Miller, H. (2010). Modeling the variability of rankings. Annals of Statistics, 38, 2562–2677.Google Scholar
  6. Inoue, K., Aki, S. (2008). Method for studying generalized birthday and coupon collection problems. Communications in Statistics—Simulation and Computation, 37, 844–862.Google Scholar
  7. Johnson, N. L., Kotz, S. (1977). Urn models and their applications. New York: Wiley.Google Scholar
  8. Kobza, J. E., Jacobson, S. H., Vaughan, D. E. (2007). A survey of the coupon collector’s problem with random sample sizes. Methodology and Computing in Applied Probability, 9, 573–584.Google Scholar
  9. Mahmoud, H. M. (2008). Pólya urn models. Boca Raton: CRC Press.CrossRefGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2012

Authors and Affiliations

  1. 1.Department of MathematicsKansai UniversitySuita-shiJapan
  2. 2.Department of MathematicsJosai UniversitySakado-shiJapan

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