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Maximizing the Expected Duration of Owning a Relatively Best Object in a Poisson Process with Rankable Observations

Part of: Sequential methods Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Aiko Kurushima  and
Katsunori Ano
Aiko Kurushima*
Affiliation:
Tokyo University of Science
Katsunori Ano*
Affiliation:
Institute of Applied Mathematics
*
∗Postal address: Department of Industrial Management and Engineering, Faculty of Engineering, Tokyo University of Science, 1-14-6 Kudan-kita, Chiyoda-ku, Tokyo, 102-0073, Japan. Email address: [email protected]
∗∗Postal address: Department of Applied Probability, Asakusabashi, Taito-ku, Tokyo, Japan. Email address: [email protected]
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Abstract

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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 Gλ(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 si(r) onwards. The value of si(r) can be obtained for each r and i as the unique root of a deterministic equation.

Keywords

Optimal stopping problemsecretary problemPoisson arrivalduration problem

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 62L15: Optimal stopping
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 2 , June 2009 , pp. 402 - 414
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Ano, K. (2000). A Poisson arrival selection problem for Gamma prior intensity with parameter r=2. In Proc. Internat. Conf. Appl. Stoch. System Modeling (Kyoto, 2000), pp. 10–17.Google Scholar
[2] Bruss, F. T. (1987). On an optimal selection problem by Cowan and Zabczyk. J. Appl. Prob. 24, 918–928.CrossRefGoogle Scholar
[3] Bruss, F. T. and Rogers, L. C. G. (1991). Pascal processes and their characterization. Stoch. Proc. Appl. 37, 331–338.CrossRefGoogle Scholar
[4] Chow, Y. S., Robbins, H. and Siegmund, D. (1971). Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston, MA.Google Scholar
[5] Cowan, R. and Zabczyk, J. (1978). An optimal selection problem associated with the Poisson process. Theory Prob. Appl. 23, 606–614.Google Scholar
[6] Ferguson, T. S. (1989). Who solved the secretary problem? Statist. Sci. 4, 282–296.Google Scholar
[7] Ferguson, T. S. (1992). Optimal Stopping and Applications. Available at http://www.math.ucla.edu/∼tom/Stopping/Contents.html.Google Scholar
[8] Ferguson, T. S., Hardwick, J. P. and Tamaki, M. (1992). Maximizing the duration of owning a relatively best object. In Strategies for Sequential Search and Selection in Real Time (Amherst, MA, (1990); Contemp. Math. 125), American Mathematical Society, Providence, RI, pp. 37–57.Google Scholar
[9] Kurushima, A. and Ano, K. (2003). A Poisson arrival selection problem for gamma prior intensity with natural number parameter. Sci. Math. Japonica 57, 217–231.Google Scholar
[10] Porosinski, A. (1987). The full-information best choice problem with a random number of observations. Stoch. Process. Appl. 24, 293–307.CrossRefGoogle Scholar
[11] Presman, È. L. and Sonin, I. M. (1972). The problem of best choice in the case of a random number of objects. Theory Prob. Appl. 17, 695–706.Google Scholar
[12] Ross, S. M. (1970). Applied Probability Models and Optimization Applications. Holden-Day, San Francisco, CA.Google Scholar
[13] Szajowski, K. (2007). A game version of the Cowan-Zabczyk-Bruss' problem. Statist. Prob. Lett. 77, 1683–1689.Google Scholar