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Optimization results for a generalized coupon collector problem

  • Emmanuelle Anceaume (a1), Yann Busnel (a2), Ernst Schulte-Geers (a3) and Bruno Sericola (a4)

Abstract

In this paper we study a generalized coupon collector problem, which consists of analyzing the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we prove that the almost uniform distribution, for which all the nonnull coupons have the same drawing probability, is the distribution which stochastically minimizes the time needed to collect a fixed number of distinct coupons. Moreover, we show that in a given closed subset of probability distributions, the distribution with all its entries, but one, equal to the smallest possible value is the one which stochastically maximizes the time needed to collect a fixed number of distinct coupons.

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* Postal address: CNRS, Campus de Beaulieu, 35042 Rennes Cedex, France.
** Postal address: Ensai, Campus de Ker-Lann, BP 37203, 35172 Bruz Cedex, France.
*** Postal address: BSI, Godesberger Allee 185-189, 53175 Bonn, Germany.
**** Postal address: Inria, Campus de Beaulieu, 35042 Rennes Cedex, France. Email address: bruno.sericola@inria.fr

References

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[1]Anceaume, E., Busnel, Y. and Sericola, B. (2015).New results on a generalized coupon collector problem using Markov chains.J. Appl. Prob. 52, 405418.
[2]Anceaume, E., Busnel, Y., Rivetti, N. and Sericola, B. (2015).Identifying global icebergs in distributed streams. In Proc. of the 34th IEEE Int. Symposium on Reliable Distributed Systems (SRDS'15), Montréal, Québec, Canada.
[3]Boneh, A. and Hofri, M. (1997).The coupon-collector problem revisited—a survey of engineering problems and computational methods.Commun. Statist. Stoch. Models 13, 3966.
[4]Doumas, A. V. and Papanicolaou, V. G. (2013).Asymptotics of the rising moments for the coupon collector's problem.Electron. J. Prob. 18, 115.
[5]Flajolet, P., Gardy, D. and Thimonier, L. (1992).Birthday paradox, coupon collectors, caching algorithms and self-organizing search.Discrete Appl. Math. 39, 207229.
[6]Marshall, A. W., Olkin, I. and Arnold, B. C. (2011).Inequalities: Theory of Majorization and Its Applications, 2nd edn.Springer, New York.
[7]Rubin, H. and Zidek, J. (1965).A waiting time distribution arising from the coupon collector's problem. Tech. Rep. 107, Department of Statistics, Stanford University.

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Optimization results for a generalized coupon collector problem

  • Emmanuelle Anceaume (a1), Yann Busnel (a2), Ernst Schulte-Geers (a3) and Bruno Sericola (a4)

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