Abstract
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.
| Language | English |
|---|---|
| Pages | 12-22 |
| Number of pages | 11 |
| Journal | Journal of Applied Probability |
| Volume | 54 |
| Issue number | 1 |
| DOIs | |
| State | Published - 2017 Mar 1 |
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Keywords
- lower bound
- Newton's inequality
- odds problem
- Optimal stopping
- secretary problem
ASJC Scopus subject areas
- Statistics and Probability
- Mathematics(all)
- Statistics, Probability and Uncertainty
Cite this
Compare the ratio of symmetric polynomials of odds to one and stop. / Matsui, Tomomi; Ano, Katsunori.
In: Journal of Applied Probability, Vol. 54, No. 1, 01.03.2017, p. 12-22.Research output: Contribution to journal › Article
}
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
VL - 54
SP - 12
EP - 22
JO - Journal of Applied Probability
T2 - Journal of Applied Probability
JF - Journal of Applied Probability
SN - 0021-9002
IS - 1
ER -