### 抄録

In this paper we deal with an optimal stopping problem whose objective is to maximize the probability of selecting k out of the last l successes, given a sequence of independent Bernoulli trials of length N, where k and l are predetermined integers satisfying 1≤k≤l<N. This problem includes some odds problems as special cases, e.g. Bruss' odds problem, Bruss and Paindaveine's problem of selecting the last l successes, and Tamaki's multiplicative odds problem for stopping at any of the last m successes. We show that an optimal stopping rule is obtained by a threshold strategy. We also present the tight lower bound and an asymptotic lower bound for the probability of a win. Interestingly, our asymptotic lower bound is attained by using a variation of the well-known secretary problem, which is a special case of the odds problem. Our approach is based on the application of Newton's inequalities and optimization technique, which gives a unified view to the previous works.

元の言語 | English |
---|---|

ページ（範囲） | 12-22 |

ページ数 | 11 |

ジャーナル | Journal of Applied Probability |

巻 | 54 |

発行部数 | 1 |

DOI | |

出版物ステータス | Published - 2017 3 1 |

### Fingerprint

### ASJC Scopus subject areas

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

### これを引用

*Journal of Applied Probability*,

*54*(1), 12-22. https://doi.org/10.1017/jpr.2016.83

**Compare the ratio of symmetric polynomials of odds to one and stop.** / Matsui, Tomomi; Ano, Katsunori.

研究成果: Article

*Journal of Applied Probability*, 巻. 54, 番号 1, pp. 12-22. https://doi.org/10.1017/jpr.2016.83

}

TY - JOUR

T1 - Compare the ratio of symmetric polynomials of odds to one and stop

AU - Matsui, Tomomi

AU - Ano, Katsunori

PY - 2017/3/1

Y1 - 2017/3/1

N2 - In this paper we deal with an optimal stopping problem whose objective is to maximize the probability of selecting k out of the last l successes, given a sequence of independent Bernoulli trials of length N, where k and l are predetermined integers satisfying 1≤k≤l<N. This problem includes some odds problems as special cases, e.g. Bruss' odds problem, Bruss and Paindaveine's problem of selecting the last l successes, and Tamaki's multiplicative odds problem for stopping at any of the last m successes. We show that an optimal stopping rule is obtained by a threshold strategy. We also present the tight lower bound and an asymptotic lower bound for the probability of a win. Interestingly, our asymptotic lower bound is attained by using a variation of the well-known secretary problem, which is a special case of the odds problem. Our approach is based on the application of Newton's inequalities and optimization technique, which gives a unified view to the previous works.

AB - In this paper we deal with an optimal stopping problem whose objective is to maximize the probability of selecting k out of the last l successes, given a sequence of independent Bernoulli trials of length N, where k and l are predetermined integers satisfying 1≤k≤l<N. This problem includes some odds problems as special cases, e.g. Bruss' odds problem, Bruss and Paindaveine's problem of selecting the last l successes, and Tamaki's multiplicative odds problem for stopping at any of the last m successes. We show that an optimal stopping rule is obtained by a threshold strategy. We also present the tight lower bound and an asymptotic lower bound for the probability of a win. Interestingly, our asymptotic lower bound is attained by using a variation of the well-known secretary problem, which is a special case of the odds problem. Our approach is based on the application of Newton's inequalities and optimization technique, which gives a unified view to the previous works.

KW - lower bound

KW - Newton's inequality

KW - odds problem

KW - Optimal stopping

KW - secretary problem

UR - http://www.scopus.com/inward/record.url?scp=85017128158&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85017128158&partnerID=8YFLogxK

U2 - 10.1017/jpr.2016.83

DO - 10.1017/jpr.2016.83

M3 - Article

AN - SCOPUS:85017128158

VL - 54

SP - 12

EP - 22

JO - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

IS - 1

ER -