### Abstract

Suppose that an unknown number of objects arrive sequentially according to a Poisson process with random intensity λ on some fixed time interval [0, T]. We assume a gamma prior density Gsλ(r. 1/a) for λ. Furthermore, we suppose that all arriving objects can be ranked uniquely among all preceding arrivals. Exactly one object can be selected. Our aim is to find a stopping time (selection time) which maximizes the time during which the selected object will stay relatively best. Our main result is the following. It is optimal to select the ith object that is relatively best and arrives at some time s ^{(r)} _{i} onwards. The value of s ^{(r)} _{i} can be obtained for each r and i as the unique root of a deterministic equation.

Original language | English |
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Pages (from-to) | 402-414 |

Number of pages | 13 |

Journal | Journal of Applied Probability |

Volume | 46 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2009 Jun |

Externally published | Yes |

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### Keywords

- Duration problem
- Optimal stopping problem
- Poisson arrival
- Secretary problem

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Journal of Applied Probability*,

*46*(2), 402-414. https://doi.org/10.1239/jap/1245676096