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A sharp lower bound for choosing the maximum of an independent sequence

Part of: Stochastic processes Sequential methods

Published online by Cambridge University Press:  09 December 2016

Pieter C. Allaart  and
José A. Islas
Pieter C. Allaart*
Affiliation:
University of North Texas
José A. Islas*
Affiliation:
University of North Texas
*
* Postal address: Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, TX 76203-5017, USA.
* Postal address: Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, TX 76203-5017, USA.
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Abstract

In this paper we consider a variation of the full-information secretary problem where the random variables to be observed are independent but not necessary identically distributed. The main result is a sharp lower bound for the optimal win probability. Precisely, if X1,...,Xn are independent random variables with known continuous distributions and Vn(X1,...,Xn):=supτℙ(Xτ=Mn), where Mn≔max{X1,...,Xn} and the supremum is over all stopping times adapted to X1,...,Xn then Vn(X1,...,Xn)≥(1-1/n)n-1, and this bound is attained. The method of proof consists in reducing the problem to that of a sequence of random variables taking at most two possible values, and then applying Bruss' sum-the-odds theorem, Bruss (2000). In order to obtain a sharp bound for each n, we improve Bruss' lower bound, Bruss (2003), for the sum-the-odds problem.

Keywords

Choosing the maximumsum-the-odds theoremstopping time

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 62L15: Optimal stopping
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 1041 - 1051
Copyright
Copyright © Applied Probability Trust 2016 

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References

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