Annals of the Institute of Statistical Mathematics

, Volume 65, Issue 3, pp 571-587

First online:

Coupon collector’s problems with statistical applications to rankings

  • Sigeo AkiAffiliated withDepartment of Mathematics, Kansai University Email author 
  • , Katuomi HiranoAffiliated withDepartment of Mathematics, Josai University

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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.


Generalized Pólya urn Dirichlet distribution Exchangeability Likelihood ratio test Permutation