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Optimal Sequential selection of n random variables under a constraint

Published online by Cambridge University Press:  14 July 2016

R. W. Chen ,
V. N. Nair ,
A. M. Odlyzko ,
L. A. Shepp  and
Y. Vardi
R. W. Chen*
Affiliation:
University of Miami
V. N. Nair*
Affiliation:
University of Miami
A. M. Odlyzko*
Affiliation:
University of Miami
L. A. Shepp*
Affiliation:
University of Miami
Y. Vardi*
Affiliation:
AT&T Bell Laboratories
*
∗Postal address: Department of Mathematics, University of Miami, Coral Gables, FL 33124, U.S.A.
∗∗Postal address: Mathematics and Statistics Research Center, AT&T Bell Laboratories, Murray Hill, NJ 07974, U.S.A.
∗∗Postal address: Mathematics and Statistics Research Center, AT&T Bell Laboratories, Murray Hill, NJ 07974, U.S.A.
∗∗Postal address: Mathematics and Statistics Research Center, AT&T Bell Laboratories, Murray Hill, NJ 07974, U.S.A.
∗∗Postal address: Mathematics and Statistics Research Center, AT&T Bell Laboratories, Murray Hill, NJ 07974, U.S.A.
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Abstract

We observe a sequence {Xk}k≧1 of i.i.d. non-negative random variables one at a time. After each observation, we select or reject the observed variable. A variable that is rejected may not be recalled. We want to select N variables as soon as possible subject to the constraint that the sum of the N selected variables does not exceed some prescribed value C > 0. In this paper, we develop a sequential selection procedure that minimizes the expected number of observed variables, and we study some of its properties. We also consider the situation where N → ∞and C/N → α > 0. Some applications are briefly discussed.

Keywords

BELLMAN EQUATIONDYNAMIC PROGRAMMINGENGINEERING APPLICATIONSSECRETARY PROBLEM
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 3 , September 1984 , pp. 537 - 547
Copyright
Copyright © Applied Probability Trust 1984 

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Footnotes

This author's research was done while consulting at Bell Laboratories.

References

[1] Campbell, G. and Samuels, S. M. (1981) Choosing the best of the current crop. Adv. Appl Prob. 15, 510–532.Google Scholar
[2] Chow, Y. S. Moriguiti, S., Robbins, H. and Samuels, S. M. (1964) Optimal selection based on relative rank. Israel J. Math. 2, 81–90.Google Scholar
[3] Chow, Y. S., Robbins, H. and Siegmund, D. (1971) Great Expectations: The Theory of Optimal Stopping. Houghton Miffin Company, Boston.Google Scholar
[4] Gilbert, J. and Mosteller, F. (1966) Recognizing the maximum of a sequence. J. Amer. Statist. Assoc. 61, 35–73.Google Scholar
[5] Mallows, C. L., Nair, V. N., Shepp, L. A. and Vardi, Y. (1983) Optimal sequential selection of secretaries.Google Scholar