### Abstract

In the subject of optimal stopping, the classical secretary problem is concerned with optimally selecting the best of n candidates when their relative ranks are observed sequentially. This problem has been extended to optimally selecting the kth best candidate for k ≥ 2. While the optimal stopping rule for k=1,2 (and all n ≥ 2) is known to be of threshold type (involving one threshold), we solve the case k=3 (and all n ≥ 3) by deriving an explicit optimal stopping rule that involves two thresholds. We also prove several inequalities for p(k, n), the maximum probability of selecting the k-th best of n candidates. It is shown that (i) p(1, n) = p(n, n) > p(k, n) for 1<k<n, (ii) p(k, n) ≥ p(k, n + 1), (iii) p(k, n) ≥ p(k + 1, n + 1) and (iv) p(k, ): = lim n→ p(k, n) is decreasing in k.

Original language | English |
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Pages (from-to) | 327-347 |

Number of pages | 21 |

Journal | Probability in the Engineering and Informational Sciences |

Volume | 33 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2019 Jul 1 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty
- Management Science and Operations Research
- Industrial and Manufacturing Engineering

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## Cite this

*Probability in the Engineering and Informational Sciences*,

*33*(3), 327-347. https://doi.org/10.1017/S0269964818000256