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Choosing the best of the current crop

Published online by Cambridge University Press:  01 July 2016

Gregory Campbell  and
Stephen M. Samuels
Gregory Campbell*
Affiliation:
Purdue University
Stephen M. Samuels*
Affiliation:
Purdue University
*
∗Postal address: Department of Statistics, Purdue University, West Lafayette, IN 47907, U.S.A.
∗Postal address: Department of Statistics, Purdue University, West Lafayette, IN 47907, U.S.A.
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Abstract

A best choice problem is presented which is intermediate between the constraints of the ‘no-information’ problem (observe only the sequence of relative ranks) and the demands of the ‘full-information’ problem (observations from a known continuous distribution). In the intermediate problem prior information is available in the form of a ‘training sample’ of size m and observations are the successive ranks of the n current items relative to their predecessors in both the current and training samples.

Optimal stopping rules for this problem depend on m and n essentially only through m + n; and, as m/(m + n) → t, their success probabilities, P*(m, n), converge rapidly to explicitly derived limits p*(t) which are the optimal success probabilities in an infinite version of the problem. For fixed n, P*(m, n) increases with m from the ‘no-information’ optimal success probability to the ‘full-information’ value for sample size n. And as t increases from 0 to 1, p*(t) increases from the ‘no-information’ limit e–1 ≍ 0·37 to the ‘full-information’ limit ≍0·58. In particular p*(0·5) ≍ 0·50.

Keywords

OPTIMAL STOPPINGRELATIVE RANKSBEST-CHOICE PROBLEMSECRETARY PROBLEM
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 3 , September 1981 , pp. 510 - 532
Copyright
Copyright © Applied Probability Trust 1981 

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References

Campbell, G. (1977) The maximum of a sequence with prior information. Purdue University Department of Statistics Mimeograph Series, No. 485.Google Scholar
Campbell, G. (1978) The secretary problem with the Dirichlet process. Inst. Math. Statist. Bull. 7, 290.Google Scholar
Chow, Y. S., Robbins, H. and Siegmund, D. (1971) Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Gianini, J. and Samuels, S. M. (1976) The infinite secretary problem. Ann. Prob. 4, 418–432.CrossRefGoogle Scholar
Gilbert, J. and Mosteller, F. (1966) Recognizing the maximum of a sequence. J. Amer. Statist. Assoc. 61, 35–73.Google Scholar
Lorenzen, T. J. (1977) Toward a more realistic formulation of the secretary problem. Purdue University Department of Statistics Mimeograph Series No. 487.Google Scholar
Mucci, A. G. (1973a) Differential equations and optimal choice problems. Ann. Statist. 1, 104–113.Google Scholar
Mucci, A. G. (1973b) On a class of secretary problems. Ann. Prob. 1, 417–427.Google Scholar
Petruccelli, J. D. (1979) Some best choice problems with partial information. Worcester Polytechnic Institute Technical Report.Google Scholar
Samuels, S. M. (1980) An explicit formula for the limiting optimal success probability in the full information best choice problem. Purdue University Statistics Department Mimeograph Series.Google Scholar
Samuels, S. M. (1981) Minimax stopping rules when the underlying distribution is uniform. J. Amer. Statist. Assoc. 76, 188–197.Google Scholar
Stewart, T. J. (1978) Optimal selection from a random sequence with learning of the underlying distribution. J. Amer. Statist. Assoc. 73, 775–780.Google Scholar