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On an optimal stopping problem of Gusein-Zade

Author

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  • Frank, Arthur Q.
  • Samuels, Stephen M.

Abstract

We study the problem of selecting one of the r best of n rankable individuals arriving in random order, in which selection must be made with a stopping rule based only on the relative ranks of the successive arrivals. For each r up to r=25, we give the limiting (as n-->[infinity]) optimal risk (probability of not selecting one of the r best) and the limiting optimal proportion of individuals to let go by before being willing to stop. (The complete limiting form of the optimal stopping rule is presented for each r up to r=10, and for r=15, 20 and 25.) We show that, for large n and r, the optical risk is approximately (1-t*)r, where t*[approximate]0.2834 is obtained as the roof of a function which is the solution to a certain differential equation. The optimal stopping rule [tau]r,n lets approximately t*n arrivals go by and then stops 'almost immediately', in the sense that [tau]r,n/n-->t* in probability as n-->[infinity], r-->[infinity]

Suggested Citation

  • Frank, Arthur Q. & Samuels, Stephen M., 1980. "On an optimal stopping problem of Gusein-Zade," Stochastic Processes and their Applications, Elsevier, vol. 10(3), pages 299-311, October.
    • Handle: RePEc:eee:spapps:v:10:y:1980:i:3:p:299-311
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    Cited by:

    1. Porosinski, Zdzislaw, 2003. "On optimal choosing of one of the k best objects," Statistics & Probability Letters, Elsevier, vol. 65(4), pages 419-432, December.
    2. Demers, Simon, 2021. "Expected duration of the no-information minimum rank problem," Statistics & Probability Letters, Elsevier, vol. 168(C).
    3. Chris Dietz & Dinard van der Laan & Ad Ridder, 2010. "Approximate Results for a Generalized Secretary Problem," Tinbergen Institute Discussion Papers 10-092/4, Tinbergen Institute.
    4. Adam Woryna, 2017. "The solution of a generalized secretary problem via analytic expressions," Journal of Combinatorial Optimization, Springer, vol. 33(4), pages 1469-1491, May.
    5. Abba M. Krieger & Ester Samuel-Cahn, 2014. "A generalized secretary problem," Discussion Paper Series dp668, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.

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