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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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