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Selecting a sequence of last successes in independent trials

Author

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  • Bruss, F. Thomas
  • Paindaveine, Davy

Abstract

Let I1, I2, . . . , In be a sequence of independent indicator functions de- fined on a probability space (Ω, A, P ). We say that index k is a success time if Ik = 1. The sequence I1, I2, . . . , In is observed sequentially. The objective of this article is to predict the l-th last success, if any, with maximum probability at the time of its occurence. We find the optimal rule and discuss briefly an algorithm to compute it in an efficient way. This generalizes the result of Bruss (1998) for l = 1, and is equivalent to the problem of (multiple) stopping with l stops on the last l successes. We extend then the model to a larger class allowing for an unknown number N of indicator functions, and present, in particular, a convenient method for an approximate solution if the success probabilities are small. We also discuss some applications of the results.

Suggested Citation

  • Bruss, F. Thomas & Paindaveine, Davy, 2000. "Selecting a sequence of last successes in independent trials," MPRA Paper 21166, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:21166
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    File URL: https://mpra.ub.uni-muenchen.de/21166/1/MPRA_paper_21166.pdf
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    References listed on IDEAS

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    1. John S. Rose, 1982. "A Problem of Optimal Choice and Assignment," Operations Research, INFORMS, vol. 30(1), pages 172-181, February.
    2. Bruss, F. Thomas, 1988. "Invariant record processes and applications to best choice modelling," Stochastic Processes and their Applications, Elsevier, vol. 30(2), pages 303-316, December.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Dendievel, Rémi, 2015. "Weber’s optimal stopping problem and generalizations," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 176-184.
    2. Gnedin, A.V.Alexander V., 2004. "Best choice from the planar Poisson process," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 317-354, June.

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    More about this item

    Keywords

    ”Sum the odds” algorithm; optimal stopping; multiple stop- ping; stopping islands; generating functions; modified secretary problems; unimodality.;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics

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