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The odds algorithm based on sequential updating and its performance

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

F. Thomas Bruss  and
Guy Louchard
F. Thomas Bruss*
Affiliation:
Université Libre de Bruxelles
Guy Louchard*
Affiliation:
Université Libre de Bruxelles
*
∗ Postal address: Université Libre de Bruxelles, Département de Mathématique, Facultés des Sciences, CP 210, Boulevard du Triomphe, B-1050 Bruxelles, Belgium. Email address: tbruss@ulb.ac.be
∗∗ Postal address: Université Libre de Bruxelles, Département d'Informatique, Facultés des Sciences, CP 212, Boulevard du Triomphe, B-1050 Bruxelles, Belgium. Email address: louchard@ulb.ac.be
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Abstract

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Let I1,I2,…,In be independent indicator functions on some probability space We suppose that these indicators can be observed sequentially. Furthermore, let T be the set of stopping times on (Ik), k=1,…,n, adapted to the increasing filtration where The odds algorithm solves the problem of finding a stopping time τ ∈ T which maximises the probability of stopping on the last Ik=1, if any. To apply the algorithm, we only need the odds for the events {Ik=1}, that is, rk=pk/(1-pk), where The goal of this paper is to offer tractable solutions for the case where the pk are unknown and must be sequentially estimated. The motivation is that this case is important for many real-world applications of optimal stopping. We study several approaches to incorporate sequential information. Our main result is a new version of the odds algorithm based on online observation and sequential updating. Questions of speed and performance of the different approaches are studied in detail, and the conclusiveness of the comparisons allows us to propose always using this algorithm to tackle selection problems of this kind.

Keywords

Optimal stoppingsequential estimationalgorithmic performancesimulationasymptotic studiesclinical trialssecretary problem

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 1 , March 2009 , pp. 131 - 153
Copyright
Copyright © Applied Probability Trust 2009 

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