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On a best-choice problem by dependent criteria

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Alexander V. Gnedin
Alexander V. Gnedin*
Affiliation:
University of Göttingen
*
∗ Postal address: Institut für Mathematische Stochastik. Universität Göttingen, Lotzestrasse 13, D-3400 Göttingen, Germany.
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Abstract

We study the problem of maximizing the probability of stopping at an object which is best in at least one of a given set of criteria, using only stopping rules based on the knowledge of whether the current object is relatively best in each of the criteria. The asymptotic results for the case of independent criteria are shown to hold in certain cases where the componentwise maxima are, pairwise, either asymptotically independent or asymptotically full dependent.

An example of the former is a random sample from a bivariate correlated normal distribution; thus our results settle a question posed recently by T. S. Ferguson.

Keywords

STOPPING RULESMULTIVARIATE EXTREMES

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60G70: Extreme value theory; extremal processes

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 1 , March 1994 , pp. 221 - 234
Copyright
Copyright © Applied Probability Trust 1994 

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Footnotes

Research supported by Deutsche Forschungsgemeinschaft.

References

[1] Berezovskii, B. A., Baryshnikov, Yu. M. and Gnedin, A. V. (1986) On a class of best choice problems. Inform. Sci. 39, 111–127.CrossRefGoogle Scholar
[2] Berezovskii, B. A., Geninson, B. A. and Rubchunskii, A. A. (1980) Optimal stopping on partially ordered objects. Automat. Remote Control 41, 1537–1542.Google Scholar
[3] Carnal, H. (1970) Die konvexe Hülle von n rotationssymmetrisch verteilten Punkten. Prob. Theory Rel. Fields 15, 168–176.Google Scholar
[4] Chow, Y. S., Robbins, H. and Sigmund, D. (1971) Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, New York.Google Scholar
[5] Feller, W. (1950) An Introduction to Probability Theory and its Applications. Wiley, New York.Google Scholar
[6] Ferguson, T. S. (1992) Best-choice problems with dependent criteria. In Strategies for Sequential Search and Selection in Real Time, pp. 135–151. Contemporary Mathematics 125, American Mathematical Society, Providence, RI.Google Scholar
[7] Gianini, J. (1977) The infinite secretary problem as the limit of the finite problem. Ann. Prob. 5, 636–644.Google Scholar
[8] Gilbert, J. and Mosteller, F. (1966) Recognizing the maximum of a sequence. J. Amer. Statist. Assoc. 61, 35–73.CrossRefGoogle Scholar
[9] Gnedin, A. V. (1981) A multicriteria problem of optimal stopping of a selection process. Automat. Remote Control 42, 981–986.Google Scholar
[10] Gnedin, A. V. (1992) Multicriteria extensions of the best choice problem: Sequential selection without linear order. In Strategies for Sequential Search and Selection in Real Time, pp. 153–172. Contemporary Mathematics 125, American Mathematical Society, Providence, RI.Google Scholar
[11] Goldie, C. M. and Resnick, S. (1989) Records in a partially ordered set. Ann. Prob. 17, 678–699.Google Scholar
[12] Renyi, A. (1962) Théorie des élements saillants d'une suite d'observations. Colloquium on Combinatorial Methods in Probability Theory, pp. 104–1115. Mathematisk Institut. Aarhus Universitet, Denmark.Google Scholar
[13] Resnick, S. (1987) Extreme Values, Regular Variation and Point Processes. Springer-Verlag, New York.Google Scholar
[14] Sakaguchi, M. (1990) Two-population secretary problems I, II. Math. Japonica 35, 917–934, 1077-1088.Google Scholar
[15] Samuels, S. M. and Chotlos, B. (1986) A multiple criteria optimal selection problem. In Adaptive Statistical Procedures and Related Topics, ed. van Ryzin, J., pp. 62–78. IMS, Hayward, CA.Google Scholar
[16] Sibuya, M. (1960) Bivariate extreme statistics. Ann. Inst. Statist. Math. 11, 195–210.Google Scholar
[17] Seneta, E. (1976) Regularly Varying Functions. Lecture Notes in Mathematics 508, Springer-Verlag, Berlin.Google Scholar
[18] Stadje, W. (1980) Efficient stopping of a random series of partially ordered points. In Multiple Criteria Decision Making Theory and Applications. Lecture Notes in Economic and Mathematical Systems 177, pp. 430–447. Springer-Verlag, Berlin.Google Scholar