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Beat the Mean: Sequential Selection by Better Than Average Rules

Part of: Nonparametric inference Limit theorems Parametric inference

Published online by Cambridge University Press:  14 July 2016

Abba M. Krieger ,
Moshe Pollak  and
Ester Samuel-Cahn
Abba M. Krieger*
Affiliation:
University of Pennsylvania
Moshe Pollak*
Affiliation:
The Hebrew University of Jerusalem
Ester Samuel-Cahn*
Affiliation:
The Hebrew University of Jerusalem
*
∗Postal address: Statistics Department, The Wharton School, University of Pennsylvania, 3730 Walnut Street, Philadelphia, PA 19104-6340, USA. Email address: krieger@wharton.upenn.edu
∗∗Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem, 91905, Israel.
∗∗Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem, 91905, Israel.
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Abstract

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We consider a sequential rule, where an item is chosen into the group, such as a university faculty member, only if his/her score is better than the average score of those already belonging to the group. We study four variables: the average score of the members of the group after k items have been selected, the time it takes (in terms of the number of observed items) to assemble a group of k items, the average score of the group after n items have been observed, and the number of items kept after the first n items have been observed. We develop the relationships between these variables, and obtain their asymptotic behavior as k (respectively, n) tends to ∞. The assumption throughout is that the items are independent and identically distributed with a continuous distribution. Though knowledge of this distribution is not needed to implement the selection rule, the asymptotic behavior does depend on the distribution. We study in some detail the exponential, Pareto, and beta distributions. Generalizations of the ‘better than average’ rule to the β better than average rules are also considered. These are rules where an item is admitted to the group only if its score is better than β times the present average of the group, where β > 0.

Keywords

Selection ruleaveragenonparametricsequential observationasymptotics

MSC classification

Secondary: 62G99: None of the above, but in this section 62F07: Ranking and selection 60F15: Strong theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008 , pp. 244 - 259
Copyright
Copyright © Applied Probability Trust 2008 

References

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Krieger, A. M., Pollak, M. and Samuel-Cahn, E. (2007a). Beat the mean: better the average. Discussion Paper 469, Center for the Study of Rationality, The Hebrew University of Jerusalem. Available at http://www.ratio.huji.ac.il/dp.php.Google Scholar
Krieger, A. M., Pollak, M. and Samuel-Cahn, E. (2007b). Select sets: rank and file. Ann. Appl. Prob. 17, 360–385.CrossRefGoogle Scholar
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